Franco Rampazzo

Impulsive control systems with commutative vector fields

We consider variational problems with control laws given by systems of ordinary differential equations whose vector fields depend linearly on the time derivativeu=(u1,...,um) of the controlu=(u1,...,um). The presence of the derivativeu, which is motivated by recent applications in Lagrangian mechanics, causes an impulsive dynamics: at any jump of the control, one expects a jump of the state.The main assumption of this paper is the commutativity of the vector fields that multiply theuα. This hypothesis allows us to associate our impulsive systems and the corresponding adjoint systems to suitable nonimpulsive control systems, to which standard techniques can be applied. In particular, we prove a maximum principle, which extends Pontryagins maximum principle to impulsive commutative systems.

Impulsive control systems without commutativity assumptions

AbstractThis paper is concerned with optimal control problems for an impulsive system of the form

Dynamic Programming for Nonlinear Systems Driven by Ordinaryand Impulsive Controls

\dot x(t) = f(t, x, u) + \sum\limits_{i = 1}^m {g_i } (t, x, u)\dot u_i ,u(t) \in U,

Nonlinear systems with unbounded controls and state constraints: a problem of proper extension

where the measurable controlu(·) is possibly discontinuous, so that the trajectories of the system must be interpreted in a generalized sense. We study in particular the case where the vector fieldsgi do not commute. By integrating the distribution generated by all the iterated Lie brackets of the vector fieldsgi, we first construct a local factorizationA1×A2 of the state space. If (x1,x2) are coordinates onA1×A2, we derive from (1) a quotient control system for the single state variablex1, withu, x2 both playing the role of controls. A density result is proved, which clarifies the relationship between the original system (1) and the quotient system. Since the quotient system turns out to be commutative, previous results valid for commutative systems can be applied, yielding existence and necessary conditions for optimal trajectories. In the final sections, two examples of impulsive systems and an application to a mechanical problem are given.

Relaxation of Control Systems Under State Constraints

Standard necessary conditions for optimal control problems with pathwise state constraints supply no useful information about minimizers in a number of cases of interest, e.g., when the left endpoint of state trajectories is fixed at x0 and x0 lies in the boundary of the state constraint set; in these cases a nonzero, but nevertheless trivial, set of multipliers exists. We give conditions for the existence of nontrivial multipliers. A feature of these conditions is that they allow nonconvex velocity sets and measurably time-dependent data. The proof techniques are based on refined estimates of the distance of a given state trajectory from the set of state trajectories satisfying the state constraint, originating in the dynamic programming literature.

A note on systems with ordinary and impulsive controls

A dynamic programming approach is considered for a class of minimum problems with impulses. The minimization domain consists of trajectories satisfying an ordinary differential equation whose right-hand side depends not only on a measurable control

Asymptotic controllability and optimal control

v

Moving Constraints as Stabilizing Controls in Classical Mechanics

but also on a second control