Emmanuel Trélat

Nonlinear Optimal Control via Occupation Measures and LMI-Relaxations

We consider the class of nonlinear optimal control problems (OCPs) with polynomial data, i.e., the differential equation, state and control constraints, and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI- (linear matrix inequality)-relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assumptions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments.

Optimal Control and Applications to Aerospace: Some Results and Challenges

This article surveys the usual techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem, in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the usual tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 1980s and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric reentry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem toward a simpler one and then of solving a series of parameterized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low-cost interplanetary space missions. The article ends with open problems and perspectives.

Uniform controllability of semidiscrete approximations of parabolic control systems

Controlling an approximation model of a controllable infinite dimensional linear control system does not necessarily yield a good approximation of the control needed for the continuous model. In the present paper, under the main assumptions that the discretized semigroup is uniformly analytic, and that the control operator is mildly unbounded, we prove that the semidiscrete approximation models are uniformly controllable. Moreover, we provide a computationally efficient way to compute the approximation controls. An example of application is implemented for the one- and two-dimensional heat equation with Neumann boundary control.

Some Properties of the Value Function and Its Level Sets for Affine Control Systems with Quadratic Cost

AbstractLet

Global Steady-State Controllability of One-Dimensional Semilinear Heat Equations

T >0

Smooth Regularization of Bang-Bang Optimal Control Problems

be fixed. We consider the optimal control problem for analytic affine systems:

GLOBAL STEADY-STATE STABILIZATION AND CONTROLLABILITY OF 1D SEMILINEAR WAVE EQUATIONS

\dot x = f_0 (x) + \sum\limits_{i = 1}^m {u_i f_i (x)}