Ilya Shvartsman

Discrete Maximum Principle for Nonsmooth Optimal Control Problems with Delays

Optimal control problems for discrete-time systems with delays are considered. Necessary optimality conditions of the discrete maximum principle type in the case of nonsmooth minimizing functions are derived. Two independent forms of the discrete maximum principle with transversality conditions described in terms of subdifferentials and superdifferentials are obtained. The superdifferential form is new even for non-delayed systems.

The Approximate Maximum Principle in Constrained Optimal Control

This paper concerns optimal control problems for dynamical systems described by a parametric family of discrete/finite-difference approximations of continuous-time control systems. Control theory for parametric systems governed by discrete approximations plays an important role in both qualitative and numerical aspects of optimal control and occupies an intermediate position in dynamic optimization: between optimal control of discrete-time (with fixed steps) and continuous-time control systems. The central result in optimal control of discrete approximation systems is the approximate maximum principle (AMP), which gives the necessary optimality condition in a perturbed maximum principle form with no a priori convexity assumptions and thus ensures the stability of the Pontryagin maximum principle (PMP) under discrete approximation procedures. The AMP has been justified for optimal control problems of smooth dynamical systems with endpoint constraints under some properness assumption imposed on the sequence of optimal controls. In this paper we show, by a series of counterexamples, that the properness assumption is essential for the validity of the AMP, and that the AMP does not hold, in its expected (lower) subdifferential form, for nonsmooth problems. Moreover, a new upper subdifferential form of the AMP is established for ordinary and time-delay control systems. The results obtained surprisingly solve (in both negative and positive directions) a long-standing and well-recognized question about the possibility of extending the AMP to nonsmooth control problems, for which the affirmative answer has been expected in the conventional lower subdifferential form.

The approximate maximum principle for constrained control systems

We consider a sequence of optimal control problems with endpoint constraints and nonsmooth terminal costs for discretized systems. The goal is to obtain necessary optimality conditions in terms of the approximate maximum principle for discrete approximations with no convexity assumptions. We construct several striking examples on the violation of the approximate maximum principle for smooth and nonsmooth problems and then establish this principle in the new super differential form for ordinary and delay systems.

Second-Order Optimality Conditions for Singular Pontryagin Local Minimizers

In this article, we use the method of finite-dimensional approximations to derive second-order necessary optimality conditions for singular Pontryagin local minimizers for an unconstrained nonlinear Mayer problem.

Second order conditions in optimal control problems with equality constraints

In this paper we consider a nonlinear optimal control problem with equality endpoint constraints. We introduce a new natural definition of singular control for which we obtain second-order necessary optimality conditions.

Necessary Optimality Conditions in Discrete Nonsmooth Optimal Control

In this paper, we provide a simple proof of the maximum principle for a nonsmooth discrete-time optimal control problem. The methodology is general and encompasses all generalized derivatives for which the Lagrange multiplier rule and the chain rule hold. This includes, but is not limited to, limiting (Mordukhovich) and Michel–Penot subdifferentials.

Nonsmooth approximate maximum principle in optimal control

In this paper we investigate stability of the Pontryagin maximum principle with respect to time discretization for optimal control problems with convex cost function and endpoint constraints. A generalization of the so-called approximate maximum principle from smooth to nonsmooth systems is obtained.

Integral control in the presence of hysteresis: an input-output approach

Using an input-output approach, it is shown that under certain mild and natural assumptions, application of integral control to the series interconnection of a hysteretic input nonlinearity, an L2-stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity guarantees tracking of constant reference signals, provided the positive time-dependent integrator gain is ultimately smaller than a certain constant determined by a positivity condition in the frequency domain. The input-output result is applied in a general state-space setting where in the linear component of the interconnection is given by a strongly stable well-posed infinite-dimensional system.

Linear programming formulation of a discrete time infinite horizon optimal control problem with time discounting criterion

This paper is devoted to a study of a discrete time infinite horizon optimal control problem with time discounting criterion. We introduce an infinite-dimensional linear programming (IDLP) problem closely related to this problem. We derive necessary and sufficient conditions of optimality for the optimal control problem in terms of the solution of the dual to this IDLP problem.

A Frailty Model in Crossover Studies with Time-to-Event Response Variable

In this article, we develop a model to study treatment, period, carryover, and other applicable effects in a crossover design with a time-to-event response variable. Because time-to-event outcomes on different treatment regimens within the crossover design are correlated for an individual, we adopt a proportional hazards frailty model. If the frailty is assumed to have a gamma distribution, and the hazard rates are piecewise constant, then the likelihood function can be determined via closed-form expressions. We illustrate the methodology via an application to a data set from an asthma clinical trial and run simulations that investigate sensitivity of the model to data generated from different distributions.