The value function of singularly perturbed control systems. (English) Zbl 0958.49019

Artstein, Z.; Gaitsgory, V.

Appl. Math. Optimization 41, No. 3, 425-445 (2000).

In this paper the authors significantly extend certain existing results concerning the asymptotic behaviour as \(\varepsilon \to 0\) of the value function \[ V_\varepsilon (t,x,y)=\inf_{u(.)} \left\{\int_t^1L(x(\tau),y(\tau),u(\tau))d\tau +\psi (x(1))\right\}, \quad (t,x,y)\in [0,1)\times \mathbb{R}^m\times \mathbb{R}^n \] where the infimum is taken over all measurable (control) functions \(u(.):[t,1]\to U\subset \mathbb{R}^l\) that “produce” the unique (absolutely continuous) solution \((x(.),y(.))\) of the problem: \[ x'(\tau)=f(x(\tau),y(\tau),u(\tau)), \;x(t)=x, \;\varepsilon y'(\tau)= g(x(\tau),y(\tau),u(\tau)), \;y(t)=y. \] Under some hypotheses on the value function itself one proves first that for any sequence \(\varepsilon_k \to 0\) there exist a subsequence, say \(\varepsilon_j\to 0\), and a “cluster function” \(V(.,.)\), such that \(V_{\varepsilon_j}(t,x,y)\to V(t,x)\) uniformly on compact subsets; next, the authors introduce the rather abstract “limit Hamiltonians”: \[ H_0(x,\lambda):=\lim_{s\to \infty}H(x,\lambda,s,y) \] \[ H(x,\lambda,s,y)=-\inf_{u(.)}\left\{ {{1}\over {s}}\int_0^s[ L(x,y(\tau), u(\tau)) +\lambda f(x,y(\tau),u(\tau))] d\tau \right\} \] where \(u(.)\) are measurable control functions and \(y(.)\) is the unique solution of the problem: \( y'(\tau)=g(x,y(\tau),u(\tau)), \;y(0)=y\) and prove (on some 5 pages) their main result, Theorem 5.3, stating that under certain hypotheses on \(V\varepsilon(.,.,.)\), \(H(.,.,.,.)\), \(H_0(.,.)\), any “cluster function” \(V(.,.)\), of \(V_\varepsilon\), is a viscosity solution of the (“limit”) Hamilton-Jacobi equation: \[ -{{\partial V}\over {\partial t}}+H_0(x, {{\partial V}\over {\partial x}})=0, \;V(1,x)=\psi (x). \] In Theorem 6.3 one identifies certain (more explicit) properties of the data that imply the rather implicit hypotheses of the main result and in a number of comments and examples the authors compare their results with previous work, in particular with those in [F. Bagagiolo and M. Bardi, SIAM J. Control Optimization 36, No. 6, 2040-2060 (1998; Zbl 0953.49031)] and [P.-L. Lions, “Generalized solutions of Hamilton-Jacobi equations” (1982; Zbl 0497.35001)], where problems “without order reduction hypothesis” are considered.

Reviewer: Stefan Mirica (Bucureşti)

MSC:

49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49L20	Dynamic programming in optimal control and differential games
34E15	Singular perturbations, general theory for ordinary differential equations

Keywords:

optimal control; singular perturbations; dynamic programming; Hamilton-Jacobi equation; viscosity solution; limit Hamiltonian