Hamilton-Jacobi equations with singular boundary conditions on a free boundary and applications to differential games. (English) Zbl 0732.35013

Bardi, Martino; Soravia, Pierpaolo

Trans. Am. Math. Soc. 325, No. 1, 205-229 (1991).

The study of the Hamilton-Jacobi equations by the form \(H(x,DU)=0\) is the object of the paper. The operator H(x,DU) is defined as \[ H(x,p):=\min_{b\in B}\max_{a\in A}\{-f(x,a,b)p-h(x,a,b)\}\text{ for all } x,p\in {\mathbb{R}}^ N, \] with A,B compact, f,h sufficiently smooth, and h satisfying the condition \(h(x,a,b)\geq h_ 0>0\) for all \(x\in {\mathbb{R}}^ N\), \(a\in A\), \(b\in B\), appearing in generalized time-optimal control problems and differential games. Two types of particular problems are considered: \[ (a)\quad H(x,DU)=0\text{ in } \Omega \setminus {\mathcal T},\quad U=g\text{ on } \partial {\mathcal T},\quad U(x)\to +\infty \quad as\quad x\to x_ 0\in \partial \Omega, \] where \({\mathcal T}\) and g are given, \({\mathcal T}\subset {\mathbb{R}}^ N\) is closed; (b) \(H(x,DU)=0\) in \({\mathcal Q}\setminus {\mathcal T}\), \(U=g\) on \(\partial {\mathcal T}\) for some open set \({\mathcal Q}\) having nonempty intersection with the closed set \({\mathcal T}.\)
In the first type of problems, the authors show that there exists at most one pair (U,\(\Omega\)) such that U is continuous in \(\Omega \setminus {\mathcal T}\) up to \(\partial {\mathcal T}\), \(\Omega \subset {\mathcal T}\) is open and the boundary value problem is satisfied in the viscosity sense. For the second type of problems they study the local unicity of the solutions and their continuity, also in the viscosity sense. A local uniqueness theorem is also given, as well as some existence results and several applications to control and game theory. In particular a relaxation theorem (weak form of the bang-bang principle) is proved for a class of nonlinear differential games.

Reviewer: St.Zanfir (Craiova)

MSC:

35F20	Nonlinear first-order PDEs
49J20	Existence theories for optimal control problems involving partial differential equations
91A23	Differential games (aspects of game theory)
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games

Keywords:

Hamilton-Jacobi equations; time-optimal control problems; differential games; viscosity; local uniqueness; existence; relaxation theorem

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