We investigate feedback control for infinite horizon optimal control problems for partial differential equations. The method is based on the coupling between Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is well-known that HJB equations suffer the so called curse of dimensionality and, therefore, a reduction of the dimension of the system is mandatory. In this repor...

The finite-horizon optimal control problem with input constraints consists in controlling the state of a dynamical system over a finite time interval (possibly unknown) minimizing a cost functional, while satisfying hard constraints on the input. For linear systems the solution of the problem often relies upon the use of bang-bang control signals. For nonlinear systems the “shape” of the optima...

We consider a robust switching control problem. The controller only observes the evolution of the state process, and thus uses feedback (closed-loop) switching strategies, a non standard class of switching controls introduced in this paper. The adverse player (nature) chooses open-loop controls that represent the so-called Knightian uncertainty, i.e., misspecifications of the model. The (half) ...

We propose multigrid methods for solving Hamilton-Jacobi-Bellman (HJB) and HamiltonJacobi-Bellman-Isaacs (HJBI) equations. The methods are based on the full approximation scheme. We propose a damped-relaxation method as smoother for multigrid. In contrast with policy iteration, the relaxation scheme is convergent for both HJB and HJBI equations. We show by local Fourier analysis that the damped...

We propose multigrid methods for solving the discrete algebraic equations arising from the discretization of the second order Hamilton–Jacobi–Bellman (HJB) and Hamilton– Jacobi–Bellman–Isaacs (HJBI) equations. We propose a damped-relaxation method as a smoother for multigrid. In contrast with the standard policy iteration, the proposed damped-relaxation scheme is convergent for both HJB and HJB...

A sufficient condition to solve an optimal control problem is to solve the Hamilton–Jacobi–Bellman (HJB) equation. However, finding a value function that satisfies the HJB equation for a nonlinear system is challenging. For an optimal control problem when a cost function is provided a priori, previous efforts have utilized feedback linearization methods which assume exact model knowledge, or ha...

An optimal control problem is considered for a stochastic differential equation with the cost functional determined by backward Volterra integral (BSVIE, short). This kind of can cover general discounting (including exponential and non-exponential) situations recursive feature. It known that such time-inconsistent in general. Therefore, instead finding global control, we look time-consistent lo...

We develop a computationally efficient learning-based forward–backward stochastic differential equations (FBSDE) controller for both continuous and hybrid dynamical (HD) systems subject to noise state constraints. Solutions optimal control (SOC) problems satisfy the Hamilton–Jacobi–Bellman (HJB) equation. Using current FBSDE-based solutions, can be obtained from HJB using deep neural networks (...

We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton–Jacobi–Bellman (HJB) equations on bounded domain $${\mathcal {O}}\subset {\mathbb {R}}^N$$ ( $$N=1,2,3$$ ) with oblique derivatives boundary conditions. These appear naturally the study optimal control diffusion processes reflection at domain. The proposed scheme is shown to s...

This paper studies the infinite-horizon optimal consumption problem with a path-dependent reference under exponential utility. The performance is measured by difference between nonnegative rate and fraction of historical maximum. running maximum process chosen as an auxiliary state process, hence value function depends on two variables. Hamilton–Jacobi–Bellman (HJB) equation can be heuristicall...