Some results on optimal control of stochastic systems with state chance constraints

Abstract : — We consider a continuous-time control problem with random initial condition and chance constraints. We solve this problem by discretizations and we prove that the discrete-time problem is convex and can be solved by the method of the logarithmic barrier function. Then we prove that when the discretization step goes to zero, the cost of the solutions of the discrete-time problems converge to the optimal cost of the continuous-time problem. I. INTRODUCTION In this paper, we consider a feedforward optimal control problem with stochastic initial state and stochastic constraints (called in many papers " chance constraints " , see [1] for an excellent review or [2] for some results related to our works). Chance constraints in optimal control problems are a powerful formalism to deal with realistic constraints. In many cases, constraints are inherently stochastic constraints, see [3]–[5]. Moreover, chance constraints are less binding than deterministic constraints and may encourage more smooth control (see our example for instance). In order to solve numerically this problem, we present a finite-dimensional stochastic problem which is closely related to the continuous time version. This problem belongs to a class of stochastic optimal control problem which is more general than the continuous-time problem but that can be solved easily with a standard method (Logarithmic Barrier Function Method – SUMT, see [6], [7]), adapted to a convex optimization problem. Therefore, we will prove that our problem is convex. One of the main originality of this work comes from the fact that we consider a degenerate normal distribution, which is not absolutely continuous w.r.t. Lebesgue measure. We show that despite this, our problem remains convex and we describe a numerical method to solve this problem. Our main objective in this work is to deal with continuous-time optimization problem. Thererfore, in the second part of this paper, we present the continuous-time version of our optimization problem and we show that the solution can be approximated by solving the discrete-time problem as explained in the first part. Let us remark that our optimal control problems are fixed final time problems. This is a suitable approach when one want to consider finite-horizon stochastic model predictive control problems ([3]–[5], [8]). In the last part, we present a short example to illustrate the method.
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Zahia Bouabbache, Eric Busvelle, Mohamed Aidene. Some results on optimal control of stochastic systems with state chance constraints. 2017 6th International Conference on Systems and Control (ICSC), May 2017, Batna, Algeria. ⟨10.1109/icosc.2017.7958705⟩. ⟨hal-01754102⟩

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