Perturbed infinite horizon optimal control problems (Q1323196)

The author studies the following infinite horizon optimal control problems \[ P(\alpha)\text{ minimize }J(x,u)= \int^ \infty_ 0 e^{- \delta t} L(x(t),u(t))dt, \] \[ \dot x(t)= f(x(t),u(t)),\quad x(0)= x_ 0+ \alpha\in \mathbb{R}^ n,\quad u(t)\in U\text{ a.e. }t\in [0,\infty), \] where \(U\subset \mathbb{R}^ m\) is a compact set, \(u\) is measurable, \(x\) is absolutely continuous and \(\alpha\) is a small parameter. The value function \(V: \mathbb{R}^ n\to \mathbb{R}\cup\{+\infty\}\) is defined by \(V(\alpha)= \inf J(x,u)\), where the infimum is taken among the class \({\mathcal F}_ \alpha\) of feasible controls. The aim of the paper is to compute \(\partial V(0)\), where \(\partial\) denotes the generalized gradient introduced by \textit{F. H. Clarke} [Optimization and nonsmooth analysis (1983; Zbl 0582.49001)]. The author also gives sufficient conditions for \(V\) to be Lipschitz near 0 and strictly differentiable at 0. The necessary conditions for optimality of problem \(P(0)\) are derived in terms of the Hamiltonian.