Control synthesis in grid schemesfor Hamilton‐Jacobi equations

A.M. Tarasyev

Annals of Operations Research, 1999, vol. 88, issue 0, 337-359

Abstract: Grid approximation schemes for constructing value functions and optimal feedbacks inproblems of guaranteed control are proposed. Value functions in optimal control problemsare usually nondifferentiable and corresponding feedbacks have a discontinuous switchingcharacter. Constructions of generalized gradients for local (convex, concave, linear) hullsare adapted to finite difference operators which approximate value functions. Optimal feedbacksare synthesized by extremal shift in the direction of generalized gradients. Bothproblems of constructing the value function and control synthesis are solved simultaneouslyin the single grid scheme. The interpolation problem is analyzed for grid values of optimalfeedbacks. Questions of correlation between spatial and temporal meshes are examined.The significance of quasiconvex properties is clarified for linear dependence of space‐timegrids. The proposed grid schemes for solving optimal guaranteed control problems can beapplied to models arising in mechanics, mathematical economics, differential and evolutionarygames. Copyright Kluwer Academic Publishers 1999

Date: 1999
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DOI: 10.1023/A:1018998817584

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