Cauchy problem for the dynamic programming equation of boundary control. (English) Zbl 0823.49018

Cannarsa, Piermarco; Tessitore, Maria Elisabetta

Zolésio, Jean-Paul (ed.), Boundary control and variation. Proceedings of the 5th working conference held in Sophia Antipolis, France, June 1992. New York, NY: Marcel Dekker, Inc.. Lect. Notes Pure Appl. Math. 163, 13-26 (1994).

The authors establish an existence and uniqueness result for the Hamilton-Jacobi equation \[ -\textstyle{{\partial\over \partial t}} v(t, x)+ H(t, x, (- A)^ \beta D_ x v(t, x))- \langle Ax+ F(x), D_ x v(t, x)\rangle= 0, v(T, x)= g(x),\tag{1} \] where \(t\in [0, T]\), \(x\in X\) and \(X\) is a real Hilbert space with scalar product \(\langle\cdot,\cdot\rangle\) and norm \(|\cdot|\). It is pointed out that equation (1) includes the dynamic programming equation of the problem: \[ u(\cdot)\text{ Inf}_ U\Biggl\{\int^ T_{t_ 0} L(s, x(s; t_ 0, x_ 0, u), u(s)) ds+ g(x(T; t_ 0, x_ 0, u))\Biggr\},\tag{2} \] where \(U\) is another Hilbert space, \(L\) and \(g\) are real-valued smooth functions and \(x(\cdot; t_ 0, x_ 0, u)\) is the mild solution of the state equation: \[ x'(t)= Ax(t)+ F(x(t))+ (- A)^ \beta Bu(t),\;x(t_ 0)= x_ 0.\tag{3} \] Under the assumption of Lipschitz continuity the authors give a definition of viscosity solution of equation (1) and obtain a comparison result between viscosity subsolution and supersolution of (1). It is also shown that the value function of problem (2)–(3) is a viscosity solution of equation (1).
For the entire collection see [Zbl 0799.00040].

Reviewer: R.N.Kaul (Delhi)

MSC:

49L20	Dynamic programming in optimal control and differential games
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games

Keywords:

value function; Hamilton-Jacobi equation; dynamic programming equation; Lipschitz continuity; viscosity solution