Lipschitzian Regularity of the Minimizing Trajectories in the Calculus of Variations and Optimal Control: a Survey (original) (raw)
Regularity properties of solutions to the basic problem in the calculus of variations
Transactions of the American Mathematical Society, 1985
This paper concerns the basic problem in the calculus of variations: minimize a functional J J defined by \[ J ( x ) = ∫ a b L ( t , x ( t ) , x ˙ ( t ) ) d t J(x) = \int _a^b {L(t,x(t),\dot x(t))\;dt} \] over a class of arcs x x whose values at a a and b b have been specified. Existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs, conditions which must be strengthened in order that the standard necessary conditions apply. The question arises: What necessary conditions hold merely under hypotheses of existence theory, say the classical Tonelli conditions? It is shown that, given a solution x x , there exists a relatively open subset Ω \Omega of [ a , b ] [a,b] , of full measure, on which x x is locally Lipschitz and satisfies a form of the Euler-Lagrange equation. The main theorem, of which this is a corollary, can also be used in conjunction with various classes of additional hypotheses to deduce the globa...
Journal of Mathematical Sciences, 2004
We study, in a unified way, the following questions related to the properties of Pontryagin extremals for optimal control problems with unrestricted controls: i) How the transformations, which define the equivalence of two problems, transform the extremals? ii) How to obtain quantities which are conserved along any extremal? iii) How to assure that the set of extremals include the minimizers predicted by the existence theory? These questions are connected to: i) the Carathéodory method which establishes a correspondence between the minimizing curves of equivalent problems; ii) the interplay between the concept of invariance and the theory of optimality conditions in optimal control, which are the concern of the theorems of Noether; iii) regularity conditions for the minimizers and the work pioneered by Tonelli.
Lipschitzian Regularity of Minimizers for Optimal Control Problems with Control-Affine Dynamics
Applied Mathematics and Optimization, 2000
We study Lagrange Problem of Optimal Control with a functional b a L (t, x (t) , u (t)) dt and control affine dynamicsẋ = f (t, x)+ g (t, x) u and (a priori) unconstrained control u ∈ IR m . We obtain conditions under which the minimizing controls of the problem are bounded -the fact which is crucial for applicability of many necessary optimality conditions, like, for example, Pontryagin Maximum Principle. As a corollary we obtain conditions for Lipschitzian regularity of minimizers of the Basic Problem of the Calculus of Variations and of the Problem of the Calculus of Variations with higher-order derivatives.
Regularity of minimizers in optimal control
2002
Abstract. We consider the Lagrange problem of optimal control with unrestricted controls–given a Lagrangian L, a dynamical equation x (t)= ϕ (t, x (t), u (t)), and boundary conditions x (a)= xa, x (b)= xb∈ Rn, find a control u (·)∈ L1 ([a, b]; Rr) such that the corresponding trajectory x (·)∈ W1, 1 ([a, b]; Rn) of the dynamical equation satisfies the boundary conditions, and the pair (x (·), u (·)) minimizes the functional J [x (·), u (·)]:=∫ ba
Foundations of the Calculus of Variations and Optimal Control
International Series in Operations Research & Management Science, 2010
In this chapter, we treat time as a continuum and derive optimality conditions for the extremization of certain functionals. We consider both variational calculus problems that are not expressed as optimal control problems and optimal control problems themselves. In this chapter, we relie on the classical notion of the variation of a functional. This classical perspective is the fastest way to obtain useful results that allow simple example problems to be solved that bolster one's understanding of continuous-time dynamic optimization.
Existence and regularity in the small in the calculus of variations
Journal of Differential Equations, 1985
A local existence theorem is proved for the basic problem in the calculus of variations, that of minimizing SL(t, X, n) dt over a class of functions x assuming given boundary conditions. The Lagrangian L is only assumed to be locally Lipschitz and strictly convex in its i variable. 8 1985 Academic PIES, IIIC. * The support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
Regularity of the Hamiltonian Along Optimal Trajectories
SIAM Journal on Control and Optimization, 2015
This paper concerns state constrained optimal control problems, in which the dynamic constraint takes the form of a differential inclusion. If the differential inclusion does not depend on time, then the Hamiltonian, evaluated along the optimal state trajectory and the costate trajectory, is independent of time. If the differential inclusion is Lipschitz continuous, then the Hamiltonian, evaluated along the optimal state trajectory and the co-state trajectory, is Lipschitz continuous. These two well-known results are examples of the following principle: the Hamiltonian, evaluated along the optimal state trajectory and the co-state trajectory, inherits the regularity properties of the differential inclusion, regarding its time dependence. We show that this principle also applies to another kind of regularity: if the differential inclusion has bounded variation w.r.t. time, then the Hamiltonian, evaluated along the optimal state trajectory and the co-state trajectory, has bounded variation. Two applications of these newly found properties are demonstrated. One is to derive improved conditions which guarantee the nondegeneracy of necessary conditions of optimality in the form of a Hamiltonian inclusion. The other application is to derive new conditions under which minimizers in the calculus of variations have bounded slope. The analysis is based on a recently proposed, local concept of differential inclusions that have bounded variation w.r.t. the time variable, in which conditions are imposed on the multifunction involved, only in a neighborhood of a given state trajectory.
Some Regularity Properties on Bolza problems in the Calculus of Variations
Comptes Rendus Mathematique, 2022
The paper summarizes the main core of the last results that we obtained in [4, 8, 17] on the regularity of the value function for a Bolza problem of a one-dimensional, vectorial problem of the calculus of variations. We are concerned with a nonautonomous Lagrangian that is possibly highly discontinuous in the state and velocity variables, nonconvex in the velocity variable and non coercive. The main results are achieved under the assumption that the Lagrangian is convex on the one-dimensional lines of the velocity variable and satisfies a local Lipschitz continuity condition w.r.t. the time variable, known in the literature as Property (S), and strictly related to the validity of the Erdmann-Du-Bois Reymond equation. Under our assumptions, there exists a minimizing sequence of Lipschitz functions. A first consequence is that we can exclude the presence of the Lavrentiev phenomenon. Moreover, under a further mild growth assumption satisfied by the minimal length functional, fully described in the paper, the above sequence may be taken with the same Lipschitz rank, even when the initial datum and initial value vary on a compact set. The Lipschitz regularity of the value function follows.