# Yuri S. LedyaevWestern Michigan University | WMU · Department of Mathematics

Yuri S. Ledyaev

1980 Ph.D. MFTI, 1990 Dr.Sc. Steklov Institute

## About

80

Publications

10,177

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3,394

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Introduction

Additional affiliations

August 1997 - present

September 1995 - August 1997

January 1995 - May 1995

Education

September 1977 - August 1980

**Moscow Institute of Physics and Technology (MFTI)**

Field of study

- Mathematics

September 1970 - May 1977

**Moscow Institute of Physics and Technology (MFTI)**

Field of study

- Mathematics

## Publications

Publications (80)

Wilcox Formula for Vector Fields on Banach Manifolds}
Yuri S.Ledyaev
We obtain an analogue of Wilcox--Snyder formula for flows of diffeomorphisms of $C^m$-smooth vector fields on infinite-dimensional Banach manifolds.
For classical linear system this formula can be efficiently used, for example, to obtain Magnus expansion of solutions.
The genera...

We consider a general calculus of variations problem with lower semicontinuous data which includes an optimal control problem with a dynamic differential inclusion constraint as a particular case. We provide a new approach to the derivation of necessary optimality conditions for such a problem, based on a new variant of the multidirectional mean-va...

Criteria for the convexity of closed sets in general Banach spaces in terms of the Clarke and Bouligand tangent cones are proved. In the case of uniformly convex spaces, these convexity criteria are stated in terms of proximal normal cones. These criteria are used to derive
sufficient conditions for the convexity of the images of convex sets under...

Criteria for the convexity of closed sets in general Banach spaces in terms of the Clarke and Bouligand tangent cones are proved. In the case of uniformly convex spaces, these convexity criteria are stated in terms of proximal normal cones. These criteria are used to derive sufficient conditions for the convexity of the images of convex sets under...

We consider a new class of problems of control under uncertainty in which uncertainty is described by a finite set of scenarios of evolution of some system parameters. We derive optimality conditions for non-anticipating strategies in terms of some nonstandard maximum principle which includes a new type of boundary value problem. These optimality c...

Dynamic optimization problems for differential inclusions on manifolds are considered. A mathematical framework for derivation of optimality conditions for generalized dynamical systems is proposed. We obtain optimality conditions in form of generalized Euler-Lagrange relations and in form of partially convexified Hamiltonian inclusions by using me...

We propose an extension of the Chronological Calculus, developed by Agrachev
and Gamkrelidze for the case of $C^\infty$-smooth dynamical systems on
finite-dimensional $C^\infty$-smooth manifolds, to the case of $C^m$-smooth
dynamical systems and infinite-dimensional $C^m$-manifolds. Due to a relaxation
in the underlying structure of the calculus, t...

We discuss a mathematical framework for analysis of optimal control problems
on infinite-dimensional manifolds. Such problems arise in study of optimization
for partial differential equations with some symmetry. It is shown that some
nonsmooth analysis methods and Lagrangian charts techniques can be used for
study of global variations of optimal tr...

Techniques of nonsmooth analysis play an important role in problems of optimization and control. It has been demonstrated that such methods provide effective and natural techniques even in problems for which the data is smooth. To this date, however, nonsmooth techniques have not been developed in the context of optimality conditions for control pr...

Envelopes or of parametric families of functions are typical non-differentiable functions arising in non-smooth analysis, optimization theory, control theory, the theory of generalized solutions of first-order partial differential equations, and other applications. In this survey formulae are obtained for sub- and supergradients of envelopes of low...

Matrix Riccati equations appear in numerous applications, especially in control engineering. In this paper we derive analytical
formulas for exact solutions of algebraic and differential matrix Riccati equations. These solutions are expressed in terms
of matrix transfer functions of appropriate linear dynamical systems.

The search for multiplier rules in dynamic optimization has been an important theme in the subject for over a century; it was central in the classical calculus of variations, and the Pontryagin maximum principle of optimal control theory is part of this quest. A more recent thread has involved problems with so-called mixed constraints involving the...

We consider a nonlinear control system which, under persistently acting disturbances, can be asymptotically driven to the
origin by some non-anticipating strategy with infinite memory (such a strategy determines a value of control u(t) at moment t using complete information on the prehistory of disturbances until moment t). We demonstrate that this...

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function. A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization proble...

This short note discusses a notion of discontinuous feedback which is appealing from a theoretical standpoint because, on the one hand, solutions of closed-loop equations always exist, and on the other hand, if a system can be stabilized in any manner whatsoever, then it can be also stabilized in our sense. Moreover, the implementation of this type...

We give a generalization of the classical Helly's theorem on intersection of convex sets in R N for the case of manifolds of nonpositive curvature. In particular, we show that if any N + 1 sets from a family of closed convex sets on N-dimensional Cartan-Hadamard manifold contain a common point, then all sets from this family contain a common point.

We give a generalization of the classical Helly's theorem on inter-section of convex sets in R N for the case of manifolds of nonpositive curvature. In particular, we show that if any N + 1 sets from a family of closed convex sets on N-dimensional Cartan-Hadamard manifold contain a common point, then all sets from this family contain a com-mon poin...

Multidirectional mean value inequalities provide estimates of the difference of the extremal value of a function on a given
bounded set and its value at a given point in terms of its (sub)-gradient at some intermediate point. A generalization of
such multidirectional mean value inequalities is derived by using new infinitesimal conditions for a wea...

The sufficient conditions for global almost-controllability of the infinite-dimensional nonlinear controllable system are given, which are similar to conditions of the classic Rashevskii-Chow theorem on connectibility for two arbitrary points of the subriemannian manifold of admissible curve. The suggested approach is based on strict invariance pro...

Nonsmooth analysis, differential analysis for functions without differentiability, has witnessed a rapid growth in the past several decades stimulated by intrinsic nonsmooth phenomena in control theory, optimization, mathematical economics and many other fields. In the past several y ears many problems in control theory, matrix analysis and geometr...

Nonsmooth analysis, differential analysis for functions without differentiability, has witnessed a rapid growth in the past several decades stimulated by intrinsic nonsmooth phenomena in control theory, optimization, mathematical economics and many other fields. In the past several years many problems in control theory, matrix analysis and geometry...

We derive conditions for generic existence and uniqueness of optimal control and trajectories for some class of finite-dimensional optimal control problems in the absence of traditional convexity assumptions. It is shown that for these problems existence and uniqueness of optimal control for a given initial point x is equivalent to the differentiab...

It is well known that various control tasks ( optimization of cost functionals, stabilization and etc.) cannot be performed by using continuous feedbacks. But an application of discontinuous feedback immediately poses a question of its robustness with respect to external disturbances, actuator and measurement errors. We discuss a concept of discont...

Optimal control emerged as a fundamental set of necessary conditions on minimizing arcs which takes account of differential constraints. The Pontryagin maximum principle directly generates the Euler equation and other classical necessary conditions from the calculus of variations, when specialized to problems without dynamic constraints. A new anal...

Given a locally defined, nondierentiable but Lipschitz Lyapunov func- tion, we construct a (discontinuous) feedback law which stabilizes the underlying system to any given tolerance. A further result shows that suitable Lyapunov functions of this type exist under mild assumptions. We also establish a robustness property of the feedback relative to...

One of the fundamental facts in control theory (Artstein's theorem) is the equivalence, for systems affine in controls, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This equivalence breaks down for general nonlinear systems, not affine in controls. One of the main results in t...

In this paper we derive optimality conditions in the form of a maximum principle for an optimal control problem for nonlinear
implicit, or descriptor, control systems. The regularity conditions which are imposed on the system allow us to reduce the
optimal control problem to an equivalent nonsmooth variational one. Nonsmooth analysis techniques tog...

An overview of recent results in nonsmooth analysis and control theory, with particular emphasis on proximal methods, is presented. The topics covered include proximal aiming and weak invariance (this being the core concept for many of the subsequent results presented), monotonicity along trajectories and Lyapunov stabilization, feedback synthesis...

Let a trajectory and control pair \((\bar x{\text{, }}\bar u{\text{)}}\) maximize globally the functional g(x(T)) in the basic optimal control problem. Then (evidently) any pair (x,u) from the level set of the functional g corresponding to the value g(\(\bar x\)(T)) is also globally optimal and satisfies the Pontryagin maximum principle. It is show...

We establish that differential inclusions corresponding to upper semicontinuous multifunctions are strongly asymptotically stable if and only if there exists a smooth Lyapunov function. Since well-known concepts of generalized solutions of differential equations with discontinuous right-hand side can be described in terms of solutions of certain re...

For systems affine in controls, Artstein's theorem (1983) provides
an equivalence, between continuous feedback stabilizability to an
equilibrium and the existence of smooth control Lyapunov functions. This
is one of the fundamental facts in nonlinear stabilization. The
equivalence breaks down for general nonlinear systems, not affine in
controls. O...

. We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit func...

The calculus of generalized gradients is the best-known and most frequently invoked part of nonsmooth analysis. Unlike proximal calculus, it can be developed in an arbitrary Banach space X. In this chapter we make a fresh start in such a setting, but this time, in contrast to Chapter 1, we begin with functions and not sets. We present the basic res...

In this chapter we study a number of different issues, each of interest in its own right. All the results obtained here build upon, and in some cases complement, those of the preceding chapters, and several of them address problems discussed in the Introduction. This is the case, for example, of the first section on constrained optimization. Some o...

Mathematics, as well as several areas of application, abounds with situations where it is desired to control the behavior of the trajectories of a given dynamical system. The goal can be either geometric (keep the state of the system in a given set, or bring it toward the set), or functional (find the trajectory that is optimal relative to a given...

We introduce in this chapter two basic constructs of nonsmooth analysis: proximal normals (to a set) and proximal subgradients (of a function). Proximal normals are direction vectors pointing outward from a set, generated by projecting a point onto the set. Proximal subgradients have a certain local support property to the epigraph of a function. I...

It is shown that every asymptotically controllable system can be
globally stabilized by means of some (discontinuous) feedback law. The
stabilizing strategy is based on pointwise optimization of a smoothed
version of a control-Lyapunov function, iteratively sending trajectories
into smaller and smaller neighborhoods of a desired equilibrium. A majo...

Consider a mappingF from a Hilbert spaceH to the subsets ofH, which is upper semicontinuous/Lipschitz, has nonconvex, noncompact values, and satisfies a linear growth condition. We give
the first necessary and sufficient conditions in this general setting for a subsetS ofH to be approximately weakly/strongly invariant with respect to approximate so...

It was shown recently by Clarke, Ledyaev, Sontag and Subbotin that any asymptotically controllable system can be stabilized by means of a certain type of discontinuous feedback. The feedback laws constructed in that work are robust with respect to actuator errors as well as to perturbations of the system dynamics. A drawback, however, is that they...

We establish regularity properties of solutions of linear quadratic optimal control problems involving state inequality constraints. Under simply stated and directly verifiable hypotheses on the data, it is shown that if the state constraint has index k > 0 then the optimal control u is k times differentiable; the kth derivative may be discontinuou...

We show how any (generalized) supersolution of the Hamilton--Jacobi equation can be used to construct a feedback pursuit strategy which guarantees (to any given tolerance) a capture time not exceeding the solution's value. If the supersolution is the value function, then a near-optimal pursuit strategy is obtained in this way. An important feature...

Results on the existence of zeros and fixed points of multifunctions in nonconvex sets are surveyed. Applications include results on the existence of equilibria in nonconvex sets which are weakly invariant with respect to a differential inclusion, as well as extensions of the classical fixed point theorems of Brouwer and Browder. The methods employ...

Proximal methods are used to determine the relationship between normal cones to a closed set in Rn and those to the closure of its complement. The geometry of outer and inner set approximations is explored as well. In the context of differential inclusions, we study the extent to which approximations and associated smoothings inherit invariance and...

4.4> xi for s(x). Let f : X Gamma! IR [ f+1g be a lower semicontinuous convex function, and let C be a nonempty closed convex set contained in the interior of the domain of f . We are interested in characterizing those x in C maximizing globally f on C, i.e. satisfying : f(x) f(x) for all x 2 C: (1) The convexity of C is not so important here since...

We present a unified approach to a complex of related issues in control theory, one based to a great extent on the methods of nonsmooth analysis. The issues include invariance, stability, equilibria, monotonicity, the Hamilton-Jacobi equation, feedback synthesis, and necessary conditions.

Concerns the question whether it be possible to find infinitesimal
conditions characterizing optimal feedback control in problems of
control under disturbance and differential games which are analogous to
Pontryagin minimum principle in optimal control. The positive answer to
this question is given in terms of some minimum principles in integral
an...

We prove a new type of mean value theorem, one in which the functional differences are estimated in multiple directions simultaneously.

We prove a new type of mean value theorem, one in which the functional differences are estimated in multiple directions simultaneously.

We establish a new mean value theorem applicable to lower semicontinuous functions on Hilbert space. A novel feature of the result is its "multidirectionality": it compares the value of a function at a point to its values on a set. We then discuss some refinements and consequences of the theorem, including applications to calculus, flow invariance,...

Necessary and sufficient conditions for viability, or weak invariance, of set-valued mapping with respect to the solutions of differential inclusion for which the right-hand side depends measurably on time are derived. The applications of these conditions to differential inequalities and to the representation of invariant solutions are considered.

It is well known that various control tasks (optimiza-tion of cost functionals, stabilization and etc.) can-not be performed by using continuous feedbacks. But an application of discontinuous feedback immediately poses a question of its robustness with respect to exter-nal disturbances, actuator and measurement errors. In this paper we discuss a co...

For problems of stabilizability and state constrained optimal control, the proximal aiming technique of nonsmooth analysis is employed in order to construct discontinuous feedback laws with respect to a generalized solution concept for the underlying dynamics. These feedbacks are universal for prescribed sets of initial data and possess robustness...

In this note, which expands the approaches of the author and E. F. Mishchenko [Sov. Math., Dokl. 33, 78-81 (1986), translation from Dokl. Akad. Nauk SSSR 286, 284-287 (1986; Zbl 0604.90145); Differ. Equations 23, 175-184 (1987), translation from Differ. Uravn. 23, No.2, 244-255 (1987; Zbl 0631.90101)], the conditions which characterize optimal stra...