Control synthesis in grid schemes for Hamilton–Jacobi equations

Grid approximation schemes for constructing value functions and optimal feedbacks in problems of guaranteed control are proposed. Value functions in optimal control problems are usually nondifferentiable and corresponding feedbacks have a discontinuous switching character. Constructions of generalized gradients for local (convex, concave, linear) hulls are adapted to finite difference operators which approximate value functions. Optimal feed-backs are synthesized by extremal shift in the direction of generalized gradients. Both problems of constructing the value function and control synthesis are solved simultaneously in the single grid scheme. The interpolation problem is analyzed for grid values of optimal feedbacks. Questions of correlation between spatial and temporal meshes are examined. The significance of quasiconvex properties is clarified for linear dependence of space–time grids. The proposed grid schemes for solving optimal guaranteed control problems can be applied to models arising in mechanics, mathematical economics, differential and evolutionary games.