On Generalized Solutions for Two Hamilton-Jacobi Equations with State Constraints
Abstract
Two Cauchy problems for Hamilton-Jacobi equation of the evolutionary type with state constraints are considered on a bounded time interval. The state space is one-dimensional. Hamiltonians of the considered problems depend on the state and momentum variables, and the dependence on the momentum variable is exponential. In the first problem, the Hamiltonian is convex in the momentum variable, and in the second problem, the Hamiltonian is concave in this variable. For the first problem, it is proved that a unique continuous viscosity solution exists, and a scheme is proposed for constructing this solution. The proposed scheme is based on the method of generalized characteristics. For the second problem, it is shown that a continuous viscosity solution does not exist, and to define a generalized solution it is necessary to specify some additional conditions
Keywords:
Hamilton-Jacobi equation, viscosity solution, non-coercive Hamiltonian, state constraints, method of characteristics, calculus of variations, Bolza problem
Downloads
References
Capuzzo-Dolcetta, I., Lions, P.-L. (1990). Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc., 318(2), 643–683.
Clarke, F. H. (1983). Tonelli's regurarity theory in the calculus of variations: Recent progress. In: Optimization and related fields. Lecture notes in mathematics (Conti, R., E. De Giorgi and F. Giannessi, eds), Vol. 1190, pp. 163–179.
Courant R., Hilbert, D. (1962). Methods of mathematical physics. Vol. 2. Partial differential equations. Interscience: NY.
Crandall, M. G., Lions, P.-L. (1983). Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., 277(1), 1–42.
Kruzhkov, S. N. (1966). Generalized solutions of nonlinear equations of the first order with several variables. I. Mat. Sb. (N. S.), 70(112), no. 3, 394–415 (in Russian).
Mirică, Ş. (1987). Generalized solutions by Cauchy's method of characteristics. Rendiconti del seminario matematico della universita di Padova, 77, 317–350.
Saakian, D. B., Rozanova, O. Akmetzhanov, A. (2008). Dynamics of the Eigen and the Crow-Kimura models for molecular evolution. Phys. Rev. E, 78(4), Art. no. 041908.
Shagalova, L. G. (2024). Continuous Generalized Solution of the Hamilton-Jacobi Equation with a Noncoercive Hamiltonian. Journal of Mathematical Sciences, 283, 487–494.
Soga, K. (2023). A remark on Tonelli's calculus of variations. Russian Journal of Nonlinear Dynamics, 19(2), 239–248.
Subbotin, A. I. (1991). Minimax Inequalities and Hamilton-Jacobi Equations. Nauka: Moscow (in Russian).
Subbotin, A. I. (1995). Generalized Solutions of First-Order PDEs. The Dynamical Optimization Perspective. Birkhäuser: Basel.
Subbotina, N. N. (2004). The method of characteristics for Hamilton–Jacobi equation and its applications in dynamical optimization. Modern Mathematics and its Applications, 20, 2955–3091.
Subbotina, N. N., Shagalova, L. G. (2011). On a solution to the Cauchy problem for the Hamilton-Jacobi equation with state constraints. Trudy Inst. Mat. i Mekh. UrO RAN, 17(2), 191–208 (in Russian).
Downloads
Published
How to Cite
Issue
Section
License
Articles of "Contributions to Game Theory and Management" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
