Section: Scientific Foundations

Optimal control and feedback control, stabilization

Optimal control.

Mathematically speaking, optimal control is the modern branch of the calculus of variations, rather well established and mature  [24] , [60] , [34] , [72] . Relying on Hamiltonian dynamics is now prevalent, instead of the standard Lagrangian formalism of the calculus of variations. Also, coming from control engineering, constraints on the control (for instance the control is a force or a torque, naturally bounded) or the state (for example in the shuttle atmospheric re-entry problem there is a constraint on the thermal flux) are imposed; the ones on the state are usual but these on the state yield more complicated necessary optimality conditions and an increased intrinsic complexity of the optimal solutions. Also, in the modern treatment, adapted numerical schemes have to be derived for effective computations of the optimal solutions.

What makes optimal control an applied field is the necessity of computing these optimal trajectories, or rather the controls that produce these trajectories (or, of course, close-by trajectories). Computing a given optimal trajectory and its control as a function of time is a demanding task, with non trivial numerical difficulties: roughly speaking, the Pontryagin Maximum Principle gives candidate optimal trajectories as solutions of a two point boundary value problem (for an ODE) which can be analyzed using mathematical tools from geometric control theory or solved numerically using shooting methods. Obtaining the optimal synthesis –the optimal control as a function of the state– is of course a more intricate problem  [34] , [37] . On the other hand the value function –the minimum of the criteria at each point– would give the solution to both questions if it were differentiable, but it is usually not, and is only a viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation (a PDE), whose study requires sophisticated non-smooth and singularity analysis.

These questions are not only academic for minimizing a cost is very relevant in many control engineering problems. However, modern engineering textbooks in nonlinear control systems like the “best-seller” [51] hardly mention optimal control, and rather put the emphasis on designing a feedback control, as regular and explicit as possible, satisfying some qualitative (and extremely important!) objectives: disturbance attenuation, decoupling, output regulation or stabilization. Optimal control is sometimes viewed as disconnected from automatic control... we shall come back to this unfortunate point.

Feedback, control Lyapunov functions, stabilization.

A control Lyapunov function (CLF) is a function that can be made a Lyapunov function (roughly speaking, a function that decreases along all trajectories, some call this an “artificial potential”) for the closed-loop system corresponding to some feedback law. This can be translated into a partial differential relation sometimes called “Artstein's (in)equation”  [27] . There is a definite parallel between a CLF for stabilization, solution of this differential inequation on the one hand, and the value function of an optimal control problem for the system, solution of a HJB equation on the other hand. Now, optimal control is a quantitative objective while stabilization is a qualitative objective; it is not surprising that Artstein (in)equation is very under-determined and has many more solutions than HJB equation, and that it may (although not always) even have smooth ones.

We have, in the team, a longstanding research record on the topic, construction of CLFs and stabilizing feedback controls. This is all the more interesting as our line of research has been pointing in almost opposite directions. [43] , [52] , [53] , [67] , [68] , [70] , [71] , [44] insist on the construction of continuous feedback, hence smooth CLFs whereas, on the contrary, [41] , [74] , [75] , [76] , [77] proceed with a very fine study of non-smooth CLFs, yet good enough (semi-concave) that they can produce a reasonable discontinuous feedback with reasonable properties.