Solutions of Nonlinear Optimal Control Problems Using Quasilinearization and Fenchel Duality

Abstract

In this paper, we consider a special class of nonlinear optimal control problems, where the control variables are box-constrained and the objective functional is strongly convex corresponding to control variables and separable with respect to the state variables and control variables. We convert solving the original nonlinear problem into solving a sequence of constrained linear-quadratic optimal control problems by quasilinearization method. In order to solve each linear-quadratic problem efficiently we turn to study its dual problem. We formulate dual problem by the scheme of Fenchel duality, the strong duality property and the saddle point property corresponding to primal and dual problem are also proved, which together ensure that solving dual problem is effective. Thus solving the sequence of control constrained linear-quadratic optimal control problems obtained by quasilinearization technique is substituted by solving the sequence of their dual problem. We solve the sequence of dual problem and obtain the solution to primal control constrained linear-quadratic problem by the saddle point property. Furthermore, the fact that solution to each subproblem finally converges to the solution to the optimality conditions of original nonlinear problem is also proved. After that we carry out numerical experiments using present approach, for each subproblem we formulate the discretized primal and dual problem by Euler discretization scheme in our experiments. Efficiency of the present method is validated by numerical results.

Constant costate iterations for finite-horizon optimal control with nonlinear dynamics

“…It is possible to backward integrate the linear differential equation ( 6), obtaining the required expression for β(x(t)), i.e., β(x(t)) = Θ ⊤ (t)g(x(t)), for any time. By relying on the definitions (3), ( 4) and (5), it is now possible to state the following theorem, which provides an iterative strategy leading to candidate optimal solutions of (1), (2).…”

“…On the other hand, since Newton method provides particularly efficient iterative strategies to tackle nonlinear optimization tasks, methods based on the linearization of the underlying plant along certain candidate trajectories have emerged with the objective of replicating Newton's arguments in an infinite-dimensional context. This can be accomplished in two different ways, see [4] and [5]. The first one consists in the linearization of the underlying dynamics, together with the expansion of the cost functional in a Taylor series up to L. Tarantino, M.Sassano and S.Galeani are with the Dipartimento di Ingegneria Civile ed Ingegneria Informatica (DICII), Università degli Studi di Roma "Tor Vergata", Via del Politecnico 1, 00133, Roma, Italy (email: lorenzo.tarantino.pvt@gmail.com, mario.sassano@uniroma2.it, sergio.galeani@uniroma2.it)…”

Finite-Horizon Optimal Control for Linear and Nonlinear Systems Relying on Constant Optimal Costate

“…On the other hand, since Newton method provides particularly efficient iterative strategies to tackle nonlinear optimization tasks, methods based on the linearization of the underlying plant along certain candidate trajectories have emerged with the objective of replicating Newton's arguments in an infinite-dimensional context. This can be accomplished in two different ways (see [20] and [21]). The first one consists in the linearization of the underlying dynamics, together with the expansion of the cost functional in a Taylor series up to second order terms, solving then a Linear Quadratic Regulator (LQR) problem at each iteration (see [22]- [25] and [26], where the latter refers to the context of stochastic systems).…”