The viability kernel algorithm for computing value functions of infinite horizon optimal control problems
The discounted infinite horizon control problem is considered, which can be characterized using the dynamics \((U,f)\), with the feedbacks \[ x'(t)= f(x(t),u(t)),\quad u(t)\in U(x(t)), \] where the state space \(X\) and the control space \(Z\) are finite-dimensional spaces, \(U:X\to Z\) associates with each state \(x\) the set \(U(x)\) of feasible controls (in general state-dependent), and \(f:\text{Graph}(U)\to X\) describes the dynamics of the system. The epigraph of the solution of such a system is characterized as a viability kernel of an auxiliary differential inclusion. The viability kernel algorithm applied to this problem provides to compute the value function of the discretized optimal control problem as the supremum of a nondecreasing sequence of functions iteratively defined. To prove the convergence of the discrete value function, the authors use the fact, that an upper Painlevé-Kuratowski limit of closed viability domains is a viability domain.
infinite horizon optimal control problems
scientific article; zbMATH DE number 930181