Method of Characteristics for Optimal Control Problems and Conservation Laws
In this paper, notions of global generalized solutions of Cauchy problems for the Hamilton–Jacobi–Bellman equation and for a quasilinear equation (a conservation law) are introduced in terms of characteristics of the Hamilton–Jacobi equation. Theorems on the existence and uniqueness of generalized solutions are proved. Representative formulas for generalized solutions are obtained and a relation between generalized solutions of the mentioned problems is justified. These results tie nonlinear scalar optimal control problems and one-dimensional stationary conservation laws.
KeywordsGeneralize Solution Cauchy Problem Optimal Control Problem Characteristic System Jacobi Equation
Unable to display preview. Download preview PDF.
- 2.L. C. Evans, Partial Differential Equations, AMS, Providence (2010).Google Scholar
- 3.N. N. Krasovskii, The Theory of Motion Control [in Russian], Nauka, Moscow (1968).Google Scholar
- 5.L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1968).Google Scholar
- 6.B. N. Pshenichnyi, “The game with simple movements and convex terminal set,” Proc. Sem.“Theory of optimal solutions,” Kiev, Inst. Cybern., 16, No. 3, 3–16 (1969).Google Scholar
- 7.B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics [in Russian], Nauka, Moscow (1968).Google Scholar
- 8.A. I. Subbotin, Generalized Solutions of First-Order Partial Differential Equations: Prospects of Dynamic Optimization [in Russian], Institute of Computer Science, Moscow–Izhevsk (2003).Google Scholar
- 9.N. N. Subbotina, “The method of characteristics for Hamilton–Jacobi equation and its applications in dynamical optimization,” Sovrem. Mat. Prilozh., 20, 2955–3091 (2004).Google Scholar