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Thematische Bibliographien / Optimal control problems involving partial differential equations / Zeitschriftenartikel

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Zeitschriftenartikel zum Thema „Optimal control problems involving partial differential equations“

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Veröffentlicht am 17. September 2021

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1

Wong, Kar Hung. „On the computational algorithms for time-lag optimal control problems“. Bulletin of the Australian Mathematical Society 32, Nr. 2 (Oktober 1985): 309–11. http://dx.doi.org/10.1017/s0004972700009989.

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In this thesis we study the following two types of hereditary optimal control problems: (i) problems governed by systems of ordinary differential equations with discrete time-delayed arguments appearing in both the state and the control variables; (ii) problems governed by parabolic partial differential equations with Neumann boundary conditions involving time-delays.

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2

Rehbock, V., S. Wang und K. L. Teo. „Computing optimal control With a hyperbolic partial differential equation“. Journal of the Australian Mathematical Society. Series B. Applied Mathematics 40, Nr. 2 (Oktober 1998): 266–87. http://dx.doi.org/10.1017/s0334270000012510.

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AbstractWe present a method for solving a class of optimal control problems involving hyperbolic partial differential equations. A numerical integration method for the solution of a general linear second-order hyperbolic partial differential equation representing the type of dynamics under consideration is given. The method, based on the piecewise bilinear finite element approximation on a rectangular mesh, is explicit. The optimal control problem is thus discretized and reduced to an ordinary optimization problem. Fast automatic differentiation is applied to calculate the exact gradient of the discretized problem so that existing optimization algorithms may be applied. Various types of constraints may be imposed on the problem. A practical application arising from the process of gas absorption is solved using the proposed method.

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3

Beyko, Ivan, Olesya Furtel und Julia Spivak. „Generalized solutions of optimal control problems“. System research and information technologies, Nr. 4 (29.12.2020): 104–14. http://dx.doi.org/10.20535/srit.2308-8893.2020.4.08.

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The problems of optimal control of systems of algebraic-integro-differential equations and partial differential equations are considered, which describe controlled processes with concentrated and distributed parameters. Generalized optimal solutions that exist for a wide range of optimal control applications are identified. Methods for constructing approximate generalized solutions are considered.

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4

de Pinho, M. do R., und R. B. Vinter. „Necessary Conditions for Optimal Control Problems Involving Nonlinear Differential Algebraic Equations“. Journal of Mathematical Analysis and Applications 212, Nr. 2 (August 1997): 493–516. http://dx.doi.org/10.1006/jmaa.1997.5523.

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5

Wong, K. H., und N. Lock. „Optimal control of a chemical reactor“. Journal of the Australian Mathematical Society. Series B. Applied Mathematics 39, Nr. 1 (Juli 1997): 61–76. http://dx.doi.org/10.1017/s0334270000009218.

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AbstractA chemical reactor problem is considered governed by partial differential equations. We wish to control the input temperature and the input oxygen concentration so that the actual output temperature can be as close to the desired output temperature as possible. By linearizing the differential equations around a nominal equation and then applying a finite-element Galerkin Scheme to the resulting system, the original problem can be converted into a sequence of linearly-constrained quadratic programming problems.

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6

Khurshudyan, Asatur Zh. „On optimal boundary and distributed control of partial integro–differential equations“. Archives of Control Sciences 24, Nr. 1 (01.03.2014): 5–25. http://dx.doi.org/10.2478/acsc-2014-0001.

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Abstract A method of optimal control problems investigation for linear partial integro-differential equations of convolution type is proposed, when control process is carried out by boundary functions and right hand side of equation. Using Fourier real generalized integral transform control problem solution is reduced to minimization procedure of chosen optimality criterion under constraints of equality type on desired control function. Optimality of control impacts is obtained for two criteria, evaluating their linear momentum and total energy. Necessary and sufficient conditions of control problem solvability are obtained for both criteria. Numerical calculations are done and control functions are plotted for both cases of control process realization.

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7

Plekhanova, Marina, und Guzel Baybulatova. „Multi-Term Fractional Degenerate Evolution Equations and Optimal Control Problems“. Mathematics 8, Nr. 4 (01.04.2020): 483. http://dx.doi.org/10.3390/math8040483.

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A theorem on unique solvability in the sense of the strong solutions is proved for a class of degenerate multi-term fractional equations in Banach spaces. It applies to the deriving of the conditions on unique solution existence for an optimal control problem to the corresponding equation. Obtained results are used to an optimal control problem study for a model system which is described by an initial-boundary value problem for a partial differential equation.

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8

Nagase, Noriaki. „On the existence of optimal control for controlled stochastic partial differential equations“. Nagoya Mathematical Journal 115 (September 1989): 73–85. http://dx.doi.org/10.1017/s0027763000001549.

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In this paper we are concerned with stochastic control problems of the following kind. Let Y(t) be a d’-dimensional Brownian motion defined on a probability space (Ω, F, Ft, P) and u(t) an admissible control. We consider the Cauchy problem of stochastic partial differential equations (SPDE in short)where L(y, u) is the 2nd order elliptic differential operator and M(y) the 1st order differential operator.

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9

Pörner, Frank, und Daniel Wachsmuth. „Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations“. Mathematical Control & Related Fields 8, Nr. 1 (2018): 315–35. http://dx.doi.org/10.3934/mcrf.2018013.

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10

Alia, Ishak. „Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach“. Mathematical Control & Related Fields 10, Nr. 4 (2020): 785–826. http://dx.doi.org/10.3934/mcrf.2020020.

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11

Nguyen, Nhan, und Mark Ardema. „Optimality of Hyperbolic Partial Differential Equations With Dynamically Constrained Periodic Boundary Control—A Flow Control Application“. Journal of Dynamic Systems, Measurement, and Control 128, Nr. 4 (26.04.2006): 946–59. http://dx.doi.org/10.1115/1.2362814.

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This paper is concerned with optimal control of a class of distributed-parameter systems governed by first-order, quasilinear hyperbolic partial differential equations that arise in optimal control problems of many physical systems such as fluids dynamics and elastodynamics. The distributed system is controlled via a forced nonlinear periodic boundary condition that describes a boundary control action. Further, the periodic boundary control is subject to a dynamic constraint imposed by a lumped-parameter system governed by ordinary differential equations that model actuator dynamics. The partial differential equations are thus coupled with the ordinary differential equations via the periodic boundary condition. Optimality of this coupled system is investigated using variational principles to seek an adjoint formulation of the optimal control problem. The results are then applied to solve a feedback control problem of the Mach number in a wind tunnel.

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12

Mustafa, Altyeb Mohammed, Zengtai Gong und Mawia Osman. „Fuzzy Optimal Control Problem of Several Variables“. Advances in Mathematical Physics 2019 (29.12.2019): 1–12. http://dx.doi.org/10.1155/2019/2182640.

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The purpose of this paper is to establish the necessary conditions for a fuzzy optimal control problem of several variables. Also, we define fuzzy optimal control problems involving isoperimetric constraints and higher order differential equations. Then, we convert these problems to fuzzy optimal control problems of several variables in order to solve these problems using the same solution method. The main results of this paper are illustrated throughout three examples, more specifically, a discussion on the strong solutions (fuzzy solutions) of our problems.

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13

Gugat, Martin, und Michael Herty. „A smoothed penalty iteration for state constrained optimal control problems for partial differential equations“. Optimization 62, Nr. 3 (März 2013): 379–95. http://dx.doi.org/10.1080/02331934.2011.588230.

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14

Harder, Felix, und Gerd Wachsmuth. „Optimality conditions for a class of inverse optimal control problems with partial differential equations“. Optimization 68, Nr. 2-3 (09.08.2018): 615–43. http://dx.doi.org/10.1080/02331934.2018.1495205.

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15

Gugat, Martin, und Michael Herty. „The smoothed-penalty algorithm for state constrained optimal control problems for partial differential equations“. Optimization Methods and Software 25, Nr. 4 (August 2010): 573–99. http://dx.doi.org/10.1080/10556780903002750.

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16

Antonietti, P. F., L. Beirão da Veiga, N. Bigoni und M. Verani. „Mimetic finite differences for nonlinear and control problems“. Mathematical Models and Methods in Applied Sciences 24, Nr. 08 (04.05.2014): 1457–93. http://dx.doi.org/10.1142/s0218202514400016.

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In this paper we review some recent applications of the mimetic finite difference method to nonlinear problems (variational inequalities and quasilinear elliptic equations) and optimal control problems governed by linear elliptic partial differential equations. Several numerical examples show the effectiveness of mimetic finite differences in building accurate numerical approximations. Finally, driven by a real-world industrial application (the numerical simulation of the extrusion process) we explore possible further applications of the mimetic finite difference method to nonlinear Stokes equations and shape optimization/free-boundary problems.

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17

Liutang, Gong, und Fei Pusheng. „Optimal control of nonsmooth system governed by quasi-linear elliptic equations“. International Journal of Mathematics and Mathematical Sciences 20, Nr. 2 (1997): 339–46. http://dx.doi.org/10.1155/s0161171297000458.

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In this paper, we discuss a class of optimal control problems of nonsmooth systems governed by quasi-linear elliptic partial differential equations, give the existence of the problem. Through the smoothness and the approximation of the original problem, we get the necessary condition, which can be considered as the Euler-Lagrange condition under quasi-linear case.

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18

Lou, Hongwei. „Existence of optimal controls for semilinear elliptic equations without Cesari-type conditions“. ANZIAM Journal 45, Nr. 1 (Juli 2003): 115–31. http://dx.doi.org/10.1017/s1446181100013183.

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AbstractOptimal control problems governed by semilinear elliptic partial differential equations are considered. No Cesari-type conditions are assumed. By proving an existence theorem and the Pontryagin maximum principle of optimal “state-control” pairs for the corresponding relaxed problems, we establish an existence theorem of optimal pairs for the original problem.

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19

Wang, Yanqing, und Zhiyong Yu. „On the partial controllability of SDEs and the exact controllability of FBSDES“. ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 68. http://dx.doi.org/10.1051/cocv/2019052.

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A notion of partial controllability (also can be called directional controllability or output controllability) is proposed for linear controlled (forward) stochastic differential equations (SDEs), which characterizes the ability of the state to reach some given random hyperplane. It generalizes the classical notion of exact controllability. For time-invariant system, checkable rank conditions ensuring SDEs’ partial controllability are provided. With some special setting, the partial controllability for SDEs is proved to be equivalent to the exact controllability for linear controlled forward-backward stochastic differential equations (FBSDEs). Moreover, we obtain some equivalent conclusions to partial controllability for SDEs or exact controllability for FBSDEs, including the validity of observability inequalities for the adjoint equations, the solvability of some optimal control problems, the solvability of norm optimal control problems, and the non-singularity of a random version of Gramian matrix.

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20

Kremsner, Stefan, Alexander Steinicke und Michaela Szölgyenyi. „A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics“. Risks 8, Nr. 4 (09.12.2020): 136. http://dx.doi.org/10.3390/risks8040136.

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In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks.

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21

Ma, Lingling, Qiuqi Li und Lijian Jiang. „Local–global model reduction method for stochastic optimal control problems constrained by partial differential equations“. Computer Methods in Applied Mechanics and Engineering 339 (September 2018): 514–41. http://dx.doi.org/10.1016/j.cma.2018.05.012.

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22

Gershom, Gossan D. Pascal, Bailly Balè und Yoro Gozo. „Optimal Control and Necessary Optimality Conditions for Nonlinear and Perturbed Dynamic Problems“. Journal of Mathematics Research 10, Nr. 6 (12.11.2018): 63. http://dx.doi.org/10.5539/jmr.v10n6p63.

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The main goal of this paper is to establish the first order necessary optimality conditions for a tumor growth model that evolves due to cancer cell proliferation. The phenomenon is modeled by a system of three-dimensional partial differential equations. We prove the existence and uniqueness of optimal control and necessary conditions of optimality are established by using the variational formulation.

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23

Darehmiraki, Majid, Mohammad Hadi Farahi und Sohrab Effati. „Solution for fractional distributed optimal control problem by hybrid meshless method“. Journal of Vibration and Control 24, Nr. 11 (13.11.2016): 2149–64. http://dx.doi.org/10.1177/1077546316678527.

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We use a hybrid local meshless method to solve the distributed optimal control problem of a system governed by parabolic partial differential equations with Caputo fractional time derivatives of order α ∈ (0, 1]. The presented meshless method is based on the linear combination of moving least squares and radial basis functions in the same compact support, this method will change between interpolation and approximation. The aim of this paper is to solve the system of coupled fractional partial differential equations, with necessary and sufficient conditions, for fractional distributed optimal control problems using a combination of moving least squares and radial basis functions. To keep matters simple, the problem has been considered in the one-dimensional case, however the techniques can be employed for both the two- and three-dimensional cases. Several test problems are employed and results of numerical experiments are presented. The obtained results confirm the acceptable accuracy of the proposed method.

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24

Zheng, Zhonghao, Xiuchun Bi und Shuguang Zhang. „Stochastic Optimization Theory of Backward Stochastic Differential Equations Driven by G-Brownian Motion“. Abstract and Applied Analysis 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/564524.

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We consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in Hu et al. (2012), we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in Zhang (2011). Then we obtain a generalized dynamic programming principle, and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.

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Manickam, K., und P. Prakash. „Mixed Finite Element Methods for Fourth Order Elliptic Optimal Control Problems“. Numerical Mathematics: Theory, Methods and Applications 9, Nr. 4 (November 2016): 528–48. http://dx.doi.org/10.4208/nmtma.2016.m1405.

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AbstractIn this paper, a priori error estimates are derived for the mixed finite element discretization of optimal control problems governed by fourth order elliptic partial differential equations. The state and co-state are discretized by Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. The error estimates derived for the state variable as well as those for the control variable seem to be new. We illustrate with a numerical example to confirm our theoretical results.

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Treanţă, Savin. „On Modified Interval-Valued Variational Control Problems with First-Order PDE Constraints“. Symmetry 12, Nr. 3 (17.03.2020): 472. http://dx.doi.org/10.3390/sym12030472.

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In this paper, a modified interval-valued variational control problem involving first-order partial differential equations (PDEs) and inequality constraints is investigated. Specifically, under some generalized convexity assumptions, we formulate and prove LU-optimality conditions for the considered interval-valued variational control problem. In order to illustrate the main results and their effectiveness, an application is provided.

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Zhou, Jianjun. „A class of infinite-horizon stochastic delay optimal control problems and a viscosity solution to the associated HJB equation“. ESAIM: Control, Optimisation and Calculus of Variations 24, Nr. 2 (26.01.2018): 639–76. http://dx.doi.org/10.1051/cocv/2017042.

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In this paper, we investigate a class of infinite-horizon optimal control problems for stochastic differential equations with delays for which the associated second order Hamilton−Jacobi−Bellman (HJB) equation is a nonlinear partial differential equation with delays. We propose a new concept for the viscosity solution including timetand identify the value function of the optimal control problems as a unique viscosity solution to the associated second order HJB equation.

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Prilepko, A. I. „Optimal control and maximum principle in (B)-spaces. Examples for partial differential equations in (H)-spaces and ordinary differential equations in Rn“. Доклады Академии наук 489, Nr. 1 (10.11.2019): 11–16. http://dx.doi.org/10.31857/s0869-5652489111-16.

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Observation and control problems in Banach (B)-spaces are investigated. On the basis of the BUME method and the monotone mapping method, a criterion of controllability and optimal controllability is formulated. The inverse controllability problem is introduced and an abstract maximum principle is formulated in (B)-spaces. For PDE in Hilbert (H)-spaces and for ODE in Rn, the integral maximum principle is proved and the optimality system is written out.

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Rasulzade, Shahla. „REQUIRED OPTIMALITY CONDITIONS IN ONE OPTIMAL CONTROL PROBLEM WITH MULTIPOINT FUNCTIONAL“. Applied Mathematics and Control Sciences, Nr. 2 (30.06.2020): 7–26. http://dx.doi.org/10.15593/2499-9873/2020.2.01.

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One specific optimal control problem with distributed parameters of the Moskalenko type with a multipoint quality functional is considered. To date, the theory of necessary first-order optimality conditions such as the Pontryagin maximum principle or its consequences has been sufficiently developed for various optimal control problems described by ordinary differential equations, i.e. for optimal control problems with lumped parameters. Many controlled processes are described by various partial differential equations (processes with distributed parameters). Some features are inherent in optimal control problems with distributed parameters, and therefore, when studying the optimal control problem with distributed parameters, in particular, when deriving various necessary optimality conditions, non-trivial difficulties arise. In particular, in the study of cases of degeneracy of the established necessary optimality conditions, fundamental difficulties arise. In the present work, we study one optimal control problem described by a system of first-order partial differential equations with a controlled initial condition under the assumption that the initial function is a solution to the Cauchy problem for ordinary differential equations. The objective function (quality criterion) is multi-point. Therefore, it becomes necessary to introduce an unconventional conjugate equation, not in differential (classical), but in integral form. In the work, using one version of the increment method, using the explicit linearization method of the original system, the necessary optimality condition is proved in the form of an analog of the maximum principle of L.S. Pontryagin. It is known that the maximum principle of L.S. Pontryagin for various optimal control problems is the strongest necessary condition for optimality. But the principle of a maximum of L.S. Pontryagin, being a necessary condition of the first order, often degenerates. Such cases are called special, and the corresponding management, special management. Based on these considerations, in the considered problem, we study the case of degeneration of the maximum principle of L.S. Pontryagin for the problem under consideration. For this purpose, a formula for incrementing the quality functional of the second order is constructed. By introducing auxiliary matrix functions, it was possible to obtain a second-order increment formula that is constructive in nature. The necessary optimality condition for special controls in the sense of the maximum principle of L.S. Pontryagin is proved. The proved necessary optimality conditions are explicit.

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Yousept, Irwin. „Optimal Control of Quasilinear $\boldsymbol{H}(\mathbf{curl})$-Elliptic Partial Differential Equations in Magnetostatic Field Problems“. SIAM Journal on Control and Optimization 51, Nr. 5 (Januar 2013): 3624–51. http://dx.doi.org/10.1137/120904299.

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31

Denkowski, Z., und S. Mortola. „Asymptotic behavior of optimal solutions to control problems for systems described by differential inclusions corresponding to partial differential equations“. Journal of Optimization Theory and Applications 78, Nr. 2 (August 1993): 365–91. http://dx.doi.org/10.1007/bf00939675.

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Wang, Yuan, Xiaochuan Luo und Sai Li. „Optimal Control Method of Parabolic Partial Differential Equations and Its Application to Heat Transfer Model in Continuous Cast Secondary Cooling Zone“. Advances in Mathematical Physics 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/585967.

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Our work is devoted to a class of optimal control problems of parabolic partial differential equations. Because of the partial differential equations constraints, it is rather difficult to solve the optimization problem. The gradient of the cost function can be found by the adjoint problem approach. Based on the adjoint problem approach, the gradient of cost function is proved to be Lipschitz continuous. An improved conjugate method is applied to solve this optimization problem and this algorithm is proved to be convergent. This method is applied to set-point values in continuous cast secondary cooling zone. Based on the real data in a plant, the simulation experiments show that the method can ensure the steel billet quality. From these experiment results, it is concluded that the improved conjugate gradient algorithm is convergent and the method is effective in optimal control problem of partial differential equations.

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Augeraud-Véron, Emmanuelle, Raouf Boucekkine und Vladimir M. Veliov. „Distributed optimal control models in environmental economics: a review“. Mathematical Modelling of Natural Phenomena 14, Nr. 1 (2019): 106. http://dx.doi.org/10.1051/mmnp/2019016.

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We review the most recent advances in distributed optimal control applied to Environmental Economics, covering in particular problems where the state dynamics are governed by partial differential equations (PDEs). This is a quite fresh application area of distributed optimal control, which has already suggested several new mathematical research lines due to the specificities of the Environmental Economics problems involved. We enhance the latter through a survey of the variety of themes and associated mathematical structures beared by this literature. We also provide a quick tour of the existing tools in the theory of distributed optimal control that have been applied so far in Environmental Economics.

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Lauriere, M., Z. Li, L. Mertz, J. Wylie und S. Zuo. „Free boundary value problems and hjb equations for the stochastic optimal control of elasto-plastic oscillators“. ESAIM: Proceedings and Surveys 65 (2019): 425–44. http://dx.doi.org/10.1051/proc/201965425.

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We consider the optimal stopping and optimal control problems related to stochastic variational inequalities modeling elasto-plastic oscillators subject to random forcing. We formally derive the corresponding free boundary value problems and Hamilton-Jacobi-Bellman equations which belong to a class of nonlinear partial of differential equations with nonlocal Dirichlet boundary conditions. Then, we focus on solving numerically these equations by employing a combination of Howard’s algorithm and the numerical approach [A backward Kolmogorov equation approach to compute means, moments and correlations of non-smooth stochastic dynamical systems; Mertz, Stadler, Wylie; 2017] for this type of boundary conditions. Numerical experiments are given.

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Akimov, Pavel A., und Vladimir N. Sidorov. „Correct Method of Analytical Solution of Multipoint Boundary Problems of Structural Analysis for Systems of Ordinary Differential Equations with Piecewise Constant Coefficients“. Advanced Materials Research 250-253 (Mai 2011): 3652–55. http://dx.doi.org/10.4028/www.scientific.net/amr.250-253.3652.

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This paper is devoted to correct method of analytical solution of multipoint boundary problems of structural analysis for systems of ordinary differential equations with piecewise constant coefficients. Its major peculiarities include universality, computer-oriented algorithm involving theory of distributions, computational stability, optimal conditionality of resultant systems and partial Jordan decomposition of matrix of coefficients, eliminating necessity of calculation of root vectors.

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Yermekkyzy, L. „VARIATIVE SOLUTION OF THE COEFFICIENT INVERSE PROBLEM FOR THE HEAT EQUATIONS“. BULLETIN Series of Physics & Mathematical Sciences 72, Nr. 4 (29.12.2020): 23–27. http://dx.doi.org/10.51889/2020-4.1728-7901.03.

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One of the main types of inverse problems for partial differential equations are problems in which the coefficients of the equations or the quantities included in them must be determined using some additional information. Such problems are called coefficient inverse problems for partial differential equations. Coefficient inverse problems (identification problems) have become the subject of close study, especially in recent years. Interest in them is caused primarily by their important applied values. They find applications in solving problems of planning the development of oil fields (determining the filtration parameters of fields), in creating new types of measuring equipment, in solving problems of environmental monitoring, etc. The standard formulation of the coefficient inverse problem contains a functional (discrepancy), physics. When formulating the statements of inverse problems, the statements of direct problems are assumed to be known. The solution to the problem is sought from the condition of its minimum. Inverse problems for partial differential equations can be posed in variational form, i.e., as optimal control problems for the corresponding systems. A variational statement of one coefficient inverse problem for a one-dimensional heat equation is considered. By the solution of the boundary value problem for each fixed control coefficient we mean a generalized solution from the Sobolev space. The questions of correctness of the considered coefficient inverse problem in the variational setting are investigated.

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Park, H. M., und O. Y. Kim. „A Reduction Method for the Boundary Control of the Heat Conduction Equation“. Journal of Dynamic Systems, Measurement, and Control 122, Nr. 3 (18.11.1998): 435–44. http://dx.doi.org/10.1115/1.1286365.

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The Karhunen–Loe`ve Galerkin procedure (Park, H. M., and Cho, D. H., 1996, “Low Dimensional Modeling of Flow Reactors,” Int. J. Heat Mass Transf., 39, pp. 3311–3323) is a type of reduction method that can be used to solve linear or nonlinear partial differential equations by reducing them to minimal sets of algebraic or ordinary differential equations. In this work, the method is used in conjunction with a conjugate gradient technique to solve the boundary optimal control problems of the heat conduction equations. It is demonstrated that the Karhunen–Loe`ve Galerkin procedure is well suited for the problems of control or optimization, where one has to solve the governing equations repeatedly but one can also estimate the approximate solution space based on the range of control variables. Choices of empirical eigenfunctions to be employed in the Karhunen–Loe`ve Galerkin procedure and issues concerning the implementations of the method are discussed. Compared to the traditional methods, the Karhunen–Loe`ve Galerkin procedure is found to solve the optimal control problems very efficiently without losing accuracy. [S0022-0434(00)00603-1]

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38

Wachsmuth, D., und J. E. Wurst. „Exponential Convergence of $hp$-Finite Element Discretization of Optimal Boundary Control Problems with Elliptic Partial Differential Equations“. SIAM Journal on Control and Optimization 54, Nr. 5 (Januar 2016): 2526–52. http://dx.doi.org/10.1137/15m1006386.

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39

Gugat, Martin. „On the turnpike property with interior decay for optimal control problems“. Mathematics of Control, Signals, and Systems 33, Nr. 2 (10.03.2021): 237–58. http://dx.doi.org/10.1007/s00498-021-00280-4.

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AbstractIn this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval $$[0,\, T]$$ [ 0 , T ] and measures the distance to a desired stationary state. In the optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough. We show that that the corresponding optimal control problems on the time intervals $$[0, \, T]$$ [ 0 , T ] give rise to a turnpike structure in the sense that for natural numbers n if T is sufficiently large, the contribution of the objective function from subintervals of [0, T] of the form $$\begin{aligned} {[}t - t/2^n,\; t + (T-t)/2^n] \end{aligned}$$ [ t - t / 2 n , t + ( T - t ) / 2 n ] is of the order $$1/\min \{t^n, (T-t)^n\}$$ 1 / min { t n , ( T - t ) n } . We also show that a similar result holds for $$\epsilon $$ ϵ -optimal solutions of the optimal control problems if $$\epsilon >0$$ ϵ > 0 is chosen sufficiently small. At the end of the paper we present both systems that are governed by ordinary differential equations and systems governed by partial differential equations where the results can be applied.

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Kazemi, Mohammad A. „A gradient technique for an optimal control problem governed by a system of nonlinear first order partial differential equations“. Journal of the Australian Mathematical Society. Series B. Applied Mathematics 36, Nr. 3 (Januar 1995): 261–73. http://dx.doi.org/10.1017/s0334270000010432.

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AbstractIn this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

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Li, Shuang, Shican Liu, Yanli Zhou, Yonghong Wu und Xiangyu Ge. „Optimal Portfolio Selection of Mean-Variance Utility with Stochastic Interest Rate“. Journal of Function Spaces 2020 (19.11.2020): 1–10. http://dx.doi.org/10.1155/2020/3153297.

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In order to tackle the problem of how investors in financial markets allocate wealth to stochastic interest rate governed by a nested stochastic differential equations (SDEs), this paper employs the Nash equilibrium theory of the subgame perfect equilibrium strategy and propose an extended Hamilton-Jacobi-Bellman (HJB) equation to analyses the optimal control over the financial system involving stochastic interest rate and state-dependent risk aversion (SDRA) mean-variance utility. By solving the corresponding nonlinear partial differential equations (PDEs) deduced from the extended HJB equation, the analytical solutions of the optimal investment strategies under time inconsistency are derived. Finally, the numerical examples provided are used to analyze how stochastic (short-term) interest rates and risk aversion affect the optimal control strategies to illustrate the validity of our results.

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Ahmad Ali, Ahmad, Michael Hinze und Heiko Kröner. „Optimal control of elliptic surface PDEs with pointwise bounds on the state“. IMA Journal of Numerical Analysis 40, Nr. 1 (29.01.2019): 226–46. http://dx.doi.org/10.1093/imanum/dry081.

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Abstract We consider a linear–quadratic optimal control problem for elliptic surface partial differential equations (PDEs) with additional state constraints. We approximate the optimization problem by a family of discrete problems and prove convergence rates for the discrete controls and the discrete states. With this we extend results known in the Euclidean setting to the surface case. We present numerical examples confirming our theoretical findings, with measures concentrated in points and measures concentrated on a line.

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Butt, Muhammad Munir. „Multigrid Method for Optimal Control Problem Constrained by Stochastic Stokes Equations with Noise“. Mathematics 9, Nr. 7 (29.03.2021): 738. http://dx.doi.org/10.3390/math9070738.

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Optimal control problems governed by stochastic partial differential equations have become an important field in applied mathematics. In this article, we investigate one such important optimization problem, that is, the stochastic Stokes control problem with forcing term perturbed by noise. A multigrid scheme with three-factor coarsening to solve the corresponding discretized control problem is presented. On staggered grids, a three-factor coarsening strategy helps in simplifying the inter-grid transfer operators and reduction in computation (CPU time). For smoothing, a distributive Gauss–Seidel scheme with a line search strategy is employed. To validate the proposed multigrid staggered grid framework, numerical results are presented with white noise at the end.

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Fung, Rong-Fong, Jyh-Horng Chou und Yu-Lung Kuo. „Optimal Boundary Control of an Axially Moving Material System“. Journal of Dynamic Systems, Measurement, and Control 124, Nr. 1 (06.02.2001): 55–61. http://dx.doi.org/10.1115/1.1435364.

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The objective of this paper is to develop an optimal boundary control strategy for the axially moving material system through a mass-damper-spring (MDS) controller at its right-hand-side (RHS) boundary. The partial differential equation (PDE) describing the axially moving material system is combined with an ordinary differential equation (ODE), which describes the MDS. The combination provides the opportunity to suppress the flexible vibration by a control force acting on the MDS. The optimal boundary control laws are designed using the output feedback method and maximum principle theory. The output feedback method only includes the states of displacement and velocity at the RHS boundary, and does not require any model discretization thereby preventing the spillover associated with discrete parameter models. By utilizing the maximum principle theory, the optimal boundary controller is expressed in terms of an adjoint variable, and the determination of the corresponding displacement and velocity is reduced to solving a set of differential equations involving the state variable, as well as the adjoint variable, subject to boundary, initial and terminal conditions. Finally, a finite difference scheme is used to validate the theoretical results.

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Yang, Feng Wei, Chandrasekhar Venkataraman, Vanessa Styles und Anotida Madzvamuse. „A Robust and Efficient Adaptive Multigrid Solver for the Optimal Control of Phase Field Formulations of Geometric Evolution Laws“. Communications in Computational Physics 21, Nr. 1 (05.12.2016): 65–92. http://dx.doi.org/10.4208/cicp.240715.080716a.

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AbstractWe propose and investigate a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws. The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth and cell motility. Despite this, many open problems remain in the analysis and approximation of such problems. In the current work we focus on a phase field formulation of the optimal control problem, hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations. Approximation of the resulting optimal control problemis computationally challenging, requiring massive amounts of computational time and memory storage. The main focus of this work is to propose, derive, implement and test an efficient solution method for such problems. The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement. An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally. Furthermore, parallelisation as well as an adaptive-step gradient update for the control are employed to further improve efficiency. Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency. A highlight of the present work is simulation results on the optimal control of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.

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Mansimov, K. B., und R. O. Mastaliyev. „Representation of the Solution of Goursat Problem for Second Order Linear Stochastic Hyperbolic Differential Equations“. Bulletin of Irkutsk State University. Series Mathematics 36 (2021): 29–43. http://dx.doi.org/10.26516/1997-7670.2021.36.29.

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The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.

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Loghin, Daniel. „Preconditioned Dirichlet-Dirichlet Methods for Optimal Control of Elliptic PDE“. Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, Nr. 2 (01.07.2018): 175–92. http://dx.doi.org/10.2478/auom-2018-0024.

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Abstract The discretization of optimal control of elliptic partial differential equations problems yields optimality conditions in the form of large sparse linear systems with block structure. Correspondingly, when the solution method is a Dirichlet-Dirichlet non-overlapping domain decomposition method, we need to solve interface problems which inherit the block structure. It is therefore natural to consider block preconditioners acting on the interface variables for the acceleration of Krylov methods with substructuring preconditioners. In this paper we describe a generic technique which employs a preconditioner block structure based on the fractional Sobolev norms corresponding to the domains of the boundary operators arising in the matrix interface problem, some of which may include a dependence on the control regularization parameter. We illustrate our approach on standard linear elliptic control problems. We present analysis which shows that the resulting iterative method converges independently of the size of the problem. We include numerical results which indicate that performance is also independent of the control regularization parameter and exhibits only a mild dependence on the number of the subdomains.

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Rafajłowicz, Ewaryst, Krystyn Styczeń und Wojciech Rafajłowicz. „A modified filter SQP method as a tool for optimal control of nonlinear systems with spatio-temporal dynamics“. International Journal of Applied Mathematics and Computer Science 22, Nr. 2 (01.06.2012): 313–26. http://dx.doi.org/10.2478/v10006-012-0023-8.

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A modified filter SQP method as a tool for optimal control of nonlinear systems with spatio-temporal dynamicsOur aim is to adapt Fletcher's filter approach to solve optimal control problems for systems described by nonlinear Partial Differential Equations (PDEs) with state constraints. To this end, we propose a number of modifications of the filter approach, which are well suited for our purposes. Then, we discuss possible ways of cooperation between the filter method and a PDE solver, and one of them is selected and tested.

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Xu, Ruimin, und Tingting Wu. „Mean-Field Backward Stochastic Evolution Equations in Hilbert Spaces and Optimal Control for BSPDEs“. Mathematical Problems in Engineering 2014 (2014): 1–18. http://dx.doi.org/10.1155/2014/718948.

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We obtain the existence and uniqueness result of the mild solutions to mean-field backward stochastic evolution equations (BSEEs) in Hilbert spaces under a weaker condition than the Lipschitz one. As an intermediate step, the existence and uniqueness result for the mild solutions of mean-field BSEEs under Lipschitz condition is also established. And then a maximum principle for optimal control problems governed by backward stochastic partial differential equations (BSPDEs) of mean-field type is presented. In this control system, the control domain need not to be convex and the coefficients, both in the state equation and in the cost functional, depend on the law of the BSPDE as well as the state and the control. Finally, a linear-quadratic optimal control problem is given to explain our theoretical results.

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Hasan, M. Mehedi, Xiangqing W. Tangpong und Om Prakash Agrawal. „Fractional optimal control of distributed systems in spherical and cylindrical coordinates“. Journal of Vibration and Control 18, Nr. 10 (13.10.2011): 1506–25. http://dx.doi.org/10.1177/1077546311408471.

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This paper presents a general formulation and numerical scheme for the fractional optimal control problem (FOCP) of distributed systems in spherical and cylindrical coordinates. The fractional derivatives are expressed in the Caputo-Sense. The performance index of FOCP is considered as a function of both the state and the control variables and the dynamic constraints are expressed by a partial fractional differential equation. A method of separation of variables is employed to separate the time and space terms, and the eigenfunction approach is used to eliminate the terms containing space parameter and define the formulation in terms of countable number of infinite state and control variables. The fractional optimal control equations are reduced to the Volterra-type integral equations. For the problems considered, only a few eigenfunctions in each direction are sufficient for the calculations to converge. The time domain is discretized into several subintervals and the result is more stable for a larger number of time segments. Various orders of fractional derivatives are analyzed and the results converge toward those of integer optimal control problems as the order approaches the integer value of 1.

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