# Applications of Dynamic Programming to Generalized Solutions for Hamilton-Jacobi Equations with State Constraints

Liubov Shagalova and Nina Subbotina^{*}

The following Cauchy problem with state constraints for Hamilton-Jacobi equation is considered: \begin{equation}\label{url} \partial u / \partial t + H(x,\partial u / \partial x)=0, \quad 0 \le t < \infty, \quad -1\le x \le 1, \end{equation} \begin{equation}\label{nu} u(0,x)=u_0(x), \quad -1\le x \le 1, \end{equation} where \begin{equation}\label{ham} H(x,p)= -f(x)+1-\frac{1+x}{2}e^{2p}-\frac{1-x}{2}e^{-2p}. \end{equation} The problem arises in Crow-Kimura model of evolution genetics. A concept of continuous generalized solutions to the problem with state constraints is suggested. The solutions are introduced with the help of viscosity and minimax solutions to auxiliary Dirichlet problems. Constructions of the generalized solutions are based on applications of dynamic programming. This approach can be considered as a generalization of the classical Cauchy method of characteristics. Properties of the characteristics are studied and applied to the constructions. The extension problem is considered for the generalized solution with known structure in a compact domain of the state space. Sufficient conditions are obtained for continuous extensions of the solution into given expanded domains in the state space. Simulation is carried out.

Mathematics Subject Classification: 49L25 93C10 34H05 49L20

Keywords: Hamilton-Jacobi equation; state constraints; generalized solutions; characteristics

Minisymposion: Dynamic Programming Approach to Optimal Control Methods and Applications