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Journal articles on the topic 'Weakly hyperbolic systems'

Author: Grafiati

Published: 25 May 2024

Last updated: 19 July 2025

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1

YONEDA, GEN, and HISA-AKI SHINKAI. "CONSTRUCTING HYPERBOLIC SYSTEMS IN THE ASHTEKAR FORMULATION OF GENERAL RELATIVITY." International Journal of Modern Physics D 09, no. 01 (2000): 13–34. http://dx.doi.org/10.1142/s0218271800000037.

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Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.

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2

Arbieto, Alexander, André Junqueira, and Bruno Santiago. "On Weakly Hyperbolic Iterated Function Systems." Bulletin of the Brazilian Mathematical Society, New Series 48, no. 1 (2016): 111–40. http://dx.doi.org/10.1007/s00574-016-0018-4.

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3

Krylovas, A., and R. Čiegis. "Asymptotic Approximation of Hyperbolic Weakly Nonlinear Systems." Journal of Nonlinear Mathematical Physics 8, no. 4 (2001): 458–70. http://dx.doi.org/10.2991/jnmp.2001.8.4.2.

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4

Spagnolo, Sergio, and Giovanni Taglialatela. "Analytic Propagation for Nonlinear Weakly Hyperbolic Systems." Communications in Partial Differential Equations 35, no. 12 (2010): 2123–63. http://dx.doi.org/10.1080/03605300903440490.

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5

Colombini, F., and Guy Métivier. "The Cauchy problem for weakly hyperbolic systems." Communications in Partial Differential Equations 43, no. 1 (2017): 25–46. http://dx.doi.org/10.1080/03605302.2017.1399906.

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6

Arbieto, Alexander, Carlos Matheus, and Maria José Pacifico. "The Bernoulli Property for Weakly Hyperbolic Systems." Journal of Statistical Physics 117, no. 1/2 (2004): 243–60. http://dx.doi.org/10.1023/b:joss.0000044058.99450.c9.

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7

D'Ancona, Piero, Tamotu Kinoshita, and Sergio Spagnolo. "Weakly hyperbolic systems with Hölder continuous coefficients." Journal of Differential Equations 203, no. 1 (2004): 64–81. http://dx.doi.org/10.1016/j.jde.2004.03.016.

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8

Souza, Rafael R. "Sub-actions for weakly hyperbolic one-dimensional systems." Dynamical Systems 18, no. 2 (2003): 165–79. http://dx.doi.org/10.1080/1468936031000136126.

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9

Alabau-Boussouira, Fatiha. "Indirect Boundary Stabilization of Weakly Coupled Hyperbolic Systems." SIAM Journal on Control and Optimization 41, no. 2 (2002): 511–41. http://dx.doi.org/10.1137/s0363012901385368.

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10

DREHER, MICHAEL, and INGO WITT. "ENERGY ESTIMATES FOR WEAKLY HYPERBOLIC SYSTEMS OF THE FIRST ORDER." Communications in Contemporary Mathematics 07, no. 06 (2005): 809–37. http://dx.doi.org/10.1142/s0219199705001969.

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For a class of first-order weakly hyperbolic pseudo-differential systems with finite time degeneracy, well-posedness of the Cauchy problem is proved in an adapted scale of Sobolev spaces. These Sobolev spaces are constructed in correspondence to the hyperbolic operator under consideration, making use of ideas from the theory of elliptic boundary value problems on manifolds with singularities. In addition, an upper bound for the loss of regularity that occurs when passing from the Cauchy data to the solutions is established. In many examples, this upper bound turns out to be sharp.

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11

DAFERMOS, C. M. "HYPERBOLIC SYSTEMS OF BALANCE LAWS WITH WEAK DISSIPATION." Journal of Hyperbolic Differential Equations 03, no. 03 (2006): 505–27. http://dx.doi.org/10.1142/s0219891606000884.

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Global BV solutions are constructed to the Cauchy problem for strictly hyperbolic systems of balance laws endowed with a rich family of entropies and source that is merely weakly dissipative, of the type induced by relaxation mechanisms.

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12

Jiang, Kai. "Local normal forms of smooth weakly hyperbolic integrable systems." Regular and Chaotic Dynamics 21, no. 1 (2016): 18–23. http://dx.doi.org/10.1134/s1560354716010020.

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13

Melo, Ítalo. "On $$\mathbb {P}$$ P -Weakly Hyperbolic Iterated Function Systems." Bulletin of the Brazilian Mathematical Society, New Series 48, no. 4 (2017): 717–32. http://dx.doi.org/10.1007/s00574-017-0042-z.

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14

Bessa, Mário, Manseob Lee, and Sandra Vaz. "Stable weakly shadowable volume-preserving systems are volume-hyperbolic." Acta Mathematica Sinica, English Series 30, no. 6 (2014): 1007–20. http://dx.doi.org/10.1007/s10114-014-3093-8.

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15

SHAO, ZHI-QIANG. "GLOBAL WEAKLY DISCONTINUOUS SOLUTIONS TO THE MIXED INITIAL–BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS." Mathematical Models and Methods in Applied Sciences 19, no. 07 (2009): 1099–138. http://dx.doi.org/10.1142/s0218202509003735.

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In this paper, we consider the mixed initial–boundary value problem for first-order quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0}. Based on the fundamental local existence results and global-in-time a priori estimates, we prove the global existence of a unique weakly discontinuous solution u = u(t, x) with small and decaying initial data, provided that each characteristic with positive velocity is weakly linearly degenerate. Some applications to quasilinear hyperbolic systems arising in physics and other disciplines, particularly to the system describing the motion of the relativistic closed string in the Minkowski space R1+n, are also given.

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16

Begun, Nikita A., Ekaterina V. Vasil’eva, Tatiana E. Zvyagintseva, and Yurii A. Iljin. "Review of the research on the qualitative theory of differential equations at St. Petersburg University. I. Stable periodic points of diffeomorphisms with homoclinic points, systems with weakly hyperbolic invariant sets." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 11, no. 2 (2024): 211–27. http://dx.doi.org/10.21638/spbu01.2024.201.

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This paper is the first in a series of review publications devoted to the results of scientific research work that has been carried out at the Department of Differential Equations of St. Petersburg University over the past 30 years. Current scientific interests of the department staff can be divided into the following directions and topics: study of stable periodic points of diffeomorphisms with homoclinic points, study of systems with weakly hyperbolic invariant sets, local qualitative theory of essentially nonlinear systems, classification of phase portraits of a family of cubic systems, stability conditions for systems with hysteretic nonlinearities and systems with nonlinearities under the generalized Routh-Hurwitz conditions (Aizerman problem). This paper presents recent results on the first two topics outlined above. The study of stable periodic points of diffeomorphisms with homoclinic points was carried out under the assumption that the stable and unstable manifolds of the hyperbolic points are tangent at a homoclinic (heteroclinic) point, and the homoclinic (heteroclinic) point is not a point with a finite order of tangency. The research of systems with weakly hyperbolic invariant sets was conducted for the case when neutral, stable, and unstable linear spaces do not satisfy the Lipschitz condition.

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17

Chen, Gui-Qiang, Wei Xiang, and Yongqian Zhang. "Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws." Communications in Partial Differential Equations 38, no. 11 (2013): 1936–70. http://dx.doi.org/10.1080/03605302.2013.828229.

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18

Krylov, A. V. "Averaging of weakly nonlinear hyperbolic systems with nonuniform integral means." Ukrainian Mathematical Journal 43, no. 5 (1991): 566–73. http://dx.doi.org/10.1007/bf01058542.

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19

Rasskazov, I. O. "The Riemann Problem for Weakly Perturbed 2 × 2 Hyperbolic Systems." Journal of Mathematical Sciences 122, no. 5 (2004): 3564–71. http://dx.doi.org/10.1023/b:joth.0000034036.97955.a8.

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20

Jiang, Ning, and C. David Levermore. "Weakly Nonlinear-Dissipative Approximations of Hyperbolic–Parabolic Systems with Entropy." Archive for Rational Mechanics and Analysis 201, no. 2 (2011): 377–412. http://dx.doi.org/10.1007/s00205-010-0361-3.

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21

Bondarev, B. V. "Averaging in hyperbolic systems subject to weakly dependent random perturbations." Ukrainian Mathematical Journal 44, no. 8 (1992): 915–23. http://dx.doi.org/10.1007/bf01057110.

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22

REULA, OSCAR A. "STRONGLY HYPERBOLIC SYSTEMS IN GENERAL RELATIVITY." Journal of Hyperbolic Differential Equations 01, no. 02 (2004): 251–69. http://dx.doi.org/10.1142/s0219891604000111.

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We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We then prove that if a system is strongly hyperbolic with respect to a given hypersurface, then it is also strongly hyperbolic with respect to any nearby surface. We then study for how much these hypersurfaces can be deformed and discuss then causality, namely what the maximal propagation speed in any given direction is. In contrast with the symmetric hyperbolic case, for which the proof of causality is geometrical and direct, relaying in energy estimates, the proof for general strongly hyperbolic systems is indirect for it is based in Holmgren's theorem. To show that the concept is needed in the area of general relativity we discuss two results for which the theory of symmetric hyperbolic systems shows to be insufficient. The first deals with the hyperbolicity analysis of systems which are second order in space derivatives; they include certain versions of the ADM and the BSSN families of equations. This analysis is considerably simplified by introducing pseudo-differential first-order evolution equations. Well-posedness for some members of the latter family systems is established by showing they satisfy the strong hyperbolicity property. Furthermore it is shown that many other systems of such families are only weakly hyperbolic, implying they should not be used for numerical modeling. The second result deals with systems having constraints. The question posed is which hyperbolicity properties, if any, are inherited from the original evolution system by the subsidiary system satisfied by the constraint quantities. The answer is that, subject to some condition on the constraints, if the evolution system is strongly hyperbolic then the subsidiary system is also strongly hyperbolic and the causality properties of both are identical.

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23

GOSSE, LAURENT, SHI JIN, and XIANTAO LI. "TWO MOMENT SYSTEMS FOR COMPUTING MULTIPHASE SEMICLASSICAL LIMITS OF THE SCHRÖDINGER EQUATION." Mathematical Models and Methods in Applied Sciences 13, no. 12 (2003): 1689–723. http://dx.doi.org/10.1142/s0218202503003082.

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Two systems of hyperbolic equations, arising in the multiphase semiclassical limit of the linear Schrödinger equations, are investigated. One stems from a Wigner measure analysis and uses a closure by the Delta functions, whereas the other relies on the classical WKB expansion and uses the Heaviside functions for closure. The two resulting moment systems are weakly and non-strictly hyperbolic respectively. They provide two different Eulerian methods able to reproduce superimposed signals with a finite number of phases. Analytical properties of these moment systems are investigated and compared. Efficient numerical discretizations and test-cases with increasing difficulty are presented.

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24

Krylovas, A., and R. Čiegis. "A REVIEW OF NUMERICAL ASYMPTOTIC AVERAGING FOR WEAKLY NONLINEAR HYPERBOLIC WAVES." Mathematical Modelling and Analysis 9, no. 3 (2004): 209–22. http://dx.doi.org/10.3846/13926292.2004.9637254.

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We present an overview of averaging method for solving weakly nonlinear hyperbolic systems. An asymptotic solution is constructed, which is uniformly valid in the “large” domain of variables t + |x| ∼ O(ϵ –1). Using this method we obtain the averaged system, which disintegrates into independent equations for the nonresonant systems. A scheme for theoretical justification of such algorithms is given and examples are presented. The averaged systems with periodic solutions are investigated for the following problems of mathematical physics: shallow water waves, gas dynamics and elastic waves. In the resonant case the averaged systems must be solved numerically. They are approximated by the finite difference schemes and the results of numerical experiments are presented.

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25

Begun, N. A. "Perturbations of weakly hyperbolic invariant sets of two-dimension periodic systems." Vestnik St. Petersburg University: Mathematics 48, no. 1 (2015): 1–8. http://dx.doi.org/10.3103/s1063454115010033.

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26

Li, Ta-Tsien, and Yue-Jun Peng. "Cauchy problem for weakly linearly degenerate hyperbolic systems in diagonal form." Nonlinear Analysis: Theory, Methods & Applications 55, no. 7-8 (2003): 937–49. http://dx.doi.org/10.1016/j.na.2003.08.010.

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27

Pesin, Ya B., and Ya G. Sinai. "Space-time chaos in the system of weakly interacting hyperbolic systems." Journal of Geometry and Physics 5, no. 3 (1988): 483–92. http://dx.doi.org/10.1016/0393-0440(88)90035-6.

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28

Rohde, Ch. "Entropy solutions for weakly coupled hyperbolic systems in several space dimensions." Zeitschrift für angewandte Mathematik und Physik 49, no. 3 (1998): 470. http://dx.doi.org/10.1007/s000000050102.

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29

Krylovas, Aleksandras. "Application of the method of stationary phase to weakly nonlinear hyperbolic systems asymptotic solving." Lietuvos matematikos rinkinys 44 (December 17, 2004): 164–68. http://dx.doi.org/10.15388/lmr.2004.31907.

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30

Krylovas, A. "ASYMPTOTIC METHOD FOR APPROXIMATION OF RESONANT INTERACTION OF NONLINEAR MULTIDIMENSIONAL HYPERBOLIC WAVES." Mathematical Modelling and Analysis 13, no. 1 (2008): 47–54. http://dx.doi.org/10.3846/1392-6292.2008.13.47-54.

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A method of averaging along characteristics of weakly nonlinear hyperbolic systems, which was presented in earlier works of the author for one dimensional waves, is generalized for some cases of multidimensional wave problems. In this work we consider such systems and discuss a way to use the internal averaging along characteristics for new problems of asymptotical integration.

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31

Alves, José, Carla Dias, Stefano Luzzatto, and Vilton Pinheiro. "SRB measures for partially hyperbolic systems whose central direction is weakly expanding." Journal of the European Mathematical Society 19, no. 10 (2017): 2911–46. http://dx.doi.org/10.4171/jems/731.

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32

Demengel, F., and J. Rauch. "Measure valued solutions of asymptotically homogeneous semilinear hyperbolic systems in one space variable." Proceedings of the Edinburgh Mathematical Society 33, no. 3 (1990): 443–60. http://dx.doi.org/10.1017/s0013091500004855.

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We study systems which in characteristic coordinates have the formwhere A is a k × k diagonal matrix with distinct real eigenvalues. The nonlinearity F is assumed to be asymptotically homogeneous in the sense, that it is a sum of two terms, one positively homogeneous of degree one in u and a second which is sublinear in u and vanishes when u = 0. In this case, F(t, x, u(t)) is meaningful provided that u(t) is a Radon measure, and, for Radon measure initial data there is a unique solution (Theorem 2.1).The main result asserts that if μn is a sequence of initial data such that, in characteristic coordinates, the positive and negative parts of each component, , converge weakly to μ±, then the solutions coverge weakly and the limit has an interesting description given by a nonlinear superposition principle.Simple weak converge of the initial data does not imply weak convergence of the solutions.

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33

Gao, Jing, and Yao-Lin Jiang. "A periodic wavelet method for the second kind of the logarithmic integral equation." Bulletin of the Australian Mathematical Society 76, no. 3 (2007): 321–36. http://dx.doi.org/10.1017/s0004972700039721.

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A periodic wavelet Galerkin method is presented in this paper to solve a weakly singular integral equations with emphasis on the second kind of Fredholm integral equations. The kernel function, which includes of a smooth part and a log weakly singular part, is discretised by the periodic Daubechies wavelets. The wavelet compression strategy and the hyperbolic cross approximation technique are used to approximate the weakly singular and smooth kernel functions. Meanwhile, the sparse matrix of systems can be correspondingly obtained. The bi-conjugate gradient iterative method is used to solve the resulting algebraic equation systems. Especially, the analytical computational formulae are presented for the log weakly singular kernel. The computational error for the representative matrix is also evaluated. The convergence rate of this algorithm is O (N-p log(N)), where p is the vanishing moment of the periodic Daubechies wavelets. Numerical experiments are provided to illustrate the correctness of the theory presented here.

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34

Barkwell, Lawrence, Peter Lancaster, and Alexander S. Markus. "Gyroscopically Stabilized Systems: A Class Of Quadratic Eigenvalue Problems With Real Spectrum." Canadian Journal of Mathematics 44, no. 1 (1991): 42–53. http://dx.doi.org/10.4153/cjm-1992-002-2.

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AbstractEigenvalue problems for selfadjoint quadratic operator polynomials L(λ) = Iλ2 + Bλ+ C on a Hilbert space H are considered where B, C∈ℒ(H), C >0, and |B| ≥ kI + k-l C for some k >0. It is shown that the spectrum of L(λ) is real. The distribution of eigenvalues on the real line and other spectral properties are also discussed. The arguments rely on the well-known theory of (weakly) hyperbolic operator polynomials.

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35

Kinoshita, Tamotu. "On the Cauchy Problem with small analytic data for nonlinear weakly hyperbolic systems." Tsukuba Journal of Mathematics 21, no. 2 (1997): 397–420. http://dx.doi.org/10.21099/tkbjm/1496163249.

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36

Fitzgibbon, W. E., and Michel Langlais. "Weakly coupled hyperbolic systems modeling the circulation of FeLV in structured feline populations." Mathematical Biosciences 165, no. 1 (2000): 79–95. http://dx.doi.org/10.1016/s0025-5564(00)00011-0.

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37

Garg, Naveen Kumar. "A class of upwind methods based on generalized eigenvectors for weakly hyperbolic systems." Numerical Algorithms 83, no. 3 (2019): 1091–121. http://dx.doi.org/10.1007/s11075-019-00717-7.

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38

Secchi, Paolo. "Anisotropic regularity of weakly stable solutions to Majda’s hyperbolic mixed problem." Journal of Hyperbolic Differential Equations 21, no. 03 (2024): 811–26. https://doi.org/10.1142/s0219891624400095.

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In this paper, we study Majda’s example for stability of mixed problems introduced in [A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53 (Springer Verlag, 1984)]. After some transformation we analyze the stability of the problem by computing the roots of the Kreiss–Lopatinskiĭ determinant and recover the different cases as in Majda [Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53 (Springer Verlag, 1984)]. Then, we focus on the weakly stable case and prove the a priori estimate of the solution. The proof follows by adapting the approach of Coulombel and Secchi [The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J. 53 (2004) 941–1012] for the linear stability of 2D compressible vortex sheets with the simplification introduced by Chen et al. [Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math. 311 (2017) 18–60]. Compared to [J.-F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J. 53 (2004) 941–1012], we improve the a priori energy estimate in that the solution is estimated in suitable weighted Sobolev spaces, anisotropic in the frequency space, whose definition reflects the properties of the associated Lopatinskiĭ determinant. Moreover, we show that this a priori energy estimate is optimal. This very simple example appears useful for the comprehension of the method of proof of the a priori estimate, beyond the technicalities of [R. M. Chen, J. Hu and D. Wang, Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math. 311 (2017) 18–60; J.-F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J. 53 (2004) 941–1012].

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39

Benzoni-Gavage, Sylvie, Frédéric Rousset, Denis Serre, and K. Zumbrun. "Generic types and transitions in hyperbolic initial–boundary-value problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 5 (2002): 1073–104. http://dx.doi.org/10.1017/s030821050000202x.

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The stability of linear initial–boundary-value problems for hyperbolic systems (with constant coefficients) is linked to the zeros of the so-called Lopatinskii determinant. Depending on the location of these zeros, problems may be either unstable, strongly stable or weakly stable. The first two classes are known to be ‘open’, in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and/or in the boundary condition.Here we show that a third open class exists, which we call ‘weakly stable of real type’. Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system.Applications to the wave equation, linear elasticity, shock waves and phase boundaries in fluid mechanics are given.

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40

Benzoni-Gavage, Sylvie, Frédéric Rousset, Denis Serre, and K. Zumbrun. "Generic types and transitions in hyperbolic initial–boundary-value problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 5 (2002): 1073–104. http://dx.doi.org/10.1017/s0308210502000537.

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The stability of linear initial–boundary-value problems for hyperbolic systems (with constant coefficients) is linked to the zeros of the so-called Lopatinskii determinant. Depending on the location of these zeros, problems may be either unstable, strongly stable or weakly stable. The first two classes are known to be ‘open’, in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and/or in the boundary condition.Here we show that a third open class exists, which we call ‘weakly stable of real type’. Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system.Applications to the wave equation, linear elasticity, shock waves and phase boundaries in fluid mechanics are given.

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41

Williams, Mark. "Weakly stable hyperbolic boundary problems with large oscillatory coefficients: Simple cascades." Journal of Hyperbolic Differential Equations 17, no. 01 (2020): 141–83. http://dx.doi.org/10.1142/s0219891620500058.

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We prove energy estimates for exact solutions to a class of linear, weakly stable, first-order hyperbolic boundary problems with “large”, oscillatory, zeroth-order coefficients, that is, coefficients whose amplitude is large, [Formula: see text], compared to the wavelength of the oscillations, [Formula: see text]. The methods that have been used previously to prove useful energy estimates for weakly stable problems with oscillatory coefficients (e.g. simultaneous diagonalization of first-order and zeroth-order parts) all appear to fail in the presence of such large coefficients. We show that our estimates provide a way to “justify geometric optics”, that is, a way to decide whether or not approximate solutions, constructed for example by geometric optics, are close to the exact solutions on a time interval independent of [Formula: see text]. Systems of this general type arise in some classical problems of “strongly nonlinear geometric optics” coming from fluid mechanics. Special assumptions that we make here do not yet allow us to treat the latter problems, but we believe the present analysis will provide some guidance on how to attack more general cases.

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42

Morisse, Baptiste. "On hyperbolicity and Gevrey well-posedness. Part three: a model of weakly hyperbolic systems." Indiana University Mathematics Journal 70, no. 2 (2021): 743–80. http://dx.doi.org/10.1512/iumj.2021.70.8198.

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43

Corli, Andrea. "Weakly non-linear geometric optics for hyperbolic systems of conservation laws with shock waves." Asymptotic Analysis 10, no. 2 (1995): 117–72. http://dx.doi.org/10.3233/asy-1995-10202.

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44

Rohde, Christian. "Upwind finite volume schemes for weakly coupled hyperbolic systems of conservation laws in 2D." Numerische Mathematik 81, no. 1 (1998): 85–123. http://dx.doi.org/10.1007/s002110050385.

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45

Margenstern, Maurice. "A Weakly Universal Cellular Automaton in the Heptagrid of the Hyperbolic Plane." Complex Systems 27, no. 4 (2018): 315–54. http://dx.doi.org/10.25088/complexsystems.27.4.315.

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46

Anikushyn, Andrii, and Khrystyna Hranishak. "Point control of linear hyperbolic integro-differential systems." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 2 (2024): 20–28. https://doi.org/10.17721/1812-5409.2024/2.4.

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Background. The work focuses on the optimal control of distributed systems described by linear hyperbolic integro-differential equations with partial derivatives and Volterra-type integral components. Such integro-differential equations are a standard subject in applied mathematics and frequently arise in studies of processes in viscoelastic media (such as amorphous polymers, semi-crystalline polymers, biopolymers, metals at very high temperatures, bituminous materials, and more). The primary goal is to prove the existence of optimal control for distributed systems modeled by these equations. Methods. We apply methods of functional analysis and the theory of distributions. The study is conducted in specially defined Hilbert spaces, where the control operator in the system's right-hand side includes generalized Dirac delta functions. We establish the main result on the existence of optimal control based on the theory of a priori estimates in negative norms, building on the foundational work of Yu. M. Berezansky and further developed by V. P. Didenko, S. I. Lyashko, and their colleagues. Results. We formulate an optimal control problem, where control of the system is governed by a control operator appearing in the right-hand side of the initial-boundary value problem. The control operator acts into spaces of generalized functions, modeling pointwise control of the system. Further, we propose appropriate Hilbert spaces for the problem's operator and the space of admissible controls. Moreover, we provide a priori estimates in negative norms, define generalized solutions, and prove the well-posedness of the initial-boundary value problem. Finally, a general theorem on the existence of optimal control is establish and the well-definedness and weak continuity of the control operator are proved. Based on these statements, we formulate a theorem on the existence of optimal control for the problem, imposing restrictions on the admissible control set and the quality criterion to be minimized. Сonclusions. We prove a theorem providing sufficient conditions for the existence of optimal control for the considered system. In particular, we demonstrate that the control operator corresponding to pointwise control is well-defined and weakly continuous from the control space to the space of right-hand sides.

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47

Korsch, Andrea, and Dietmar Kröner. "On existence and uniqueness of entropy solutions of weakly coupled hyperbolic systems on evolving surfaces." Computers & Fluids 169 (June 2018): 296–308. http://dx.doi.org/10.1016/j.compfluid.2017.08.021.

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Qu, Peng, and Cunming Liu. "Global classical solutions to partially dissipative quasilinear hyperbolic systems with one weakly linearly degenerate characteristic." Chinese Annals of Mathematics, Series B 33, no. 3 (2012): 333–50. http://dx.doi.org/10.1007/s11401-012-0715-2.

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Dumas, E. "Nonlinear diffractive optics with curved phases: beam dispersion and transition between light and shadow." Asymptotic Analysis 38, no. 1 (2004): 47–91. https://doi.org/10.3233/asy-2004-615.

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Abstract:

We give asymptotic descriptions of smooth oscillating solutions of hyperbolic systems with variable coefficients, in the weakly nonlinear diffractive optics regime. The dependence of the coefficients of the system in the space–time variable (corresponding to propagation in a non‐homogeneous medium) implies that the rays are not parallel lines – the same occurs with non‐planar initial phases. Approximations are given by WKB asymptotics with 3‐scales profiles and curved phases. The fastest scale concerns oscillations, while the slowest one describes the modulation of the envelope, which is along rays for the oscillatory components. We consider two kinds of behaviors at the intermediate scale: ‘weakly decaying’ (Sobolev), giving the transverse evolution of a ‘ray packet’, and ‘shock‐type’ profiles describing a region of rapid transition for the amplitude.

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Benoit, Antoine. "WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: Time depending loss of derivatives." Journal of Hyperbolic Differential Equations 18, no. 03 (2021): 557–608. http://dx.doi.org/10.1142/s0219891621500181.

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Abstract:

We are interested in geometric optics expansions for linear hyperbolic systems of equations defined in the strip [Formula: see text]. More precisely the aim of this paper is to describe the influence of the boundary conditions on the behavior of the solution. This question has already been addressed in [A. Benoit, Wkb expansions for hyperbolic boundary value problems in a strip: Selfinteraction meets strong well-posedness, J. Inst. Math. Jussieu 19(5) (2020) 1629–1675] for stable boundary conditions. Here we do not require that the boundary conditions lead to strongly well-posed problems but only to weakly well-posed problems (that is loss(es) of derivatives are possible). The question is thus to determine what can be the minimal loss of derivatives in the energy estimate of the solution. The main result of this paper is to show, thanks to geometric optics expansions, that if the strip problem admits a boundary in the so-called [Formula: see text]-class of [S. Benzoni-Gavage, F. Rousset, D. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 132(5) (2002) 1073–1104] then the loss of derivatives shall be at least increasing with the time of resolution. More precisely this loss is bounded by below by a step function increasing with respect to time which depends on the minimal time needed to perform a full regeneration of the wave packet.

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