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Our contribution deals with a class of Optimal Control Problems (OCPs) of dynamic systems with randomly varying time delays. We study the minimax-type OCPs associated with a family of delayed differential equations. The presented minimax dynamic optimization has a natural interpretation as a robustness (in optimization) with respect to the possible delays in control system under consideration. A specific structure of a delayed model makes it possible to reduce the originally given sophisticated OCP to an equivalent convex program in an Euclidean space. This analytic transformation implies a possibility to derive the necessary and sufficient optimality conditions for the original OCP. Moreover, it also allows consideration of the wide range of effective numerical procedures for the constructive treatment of the obtained convex-like OCP. The concrete computational methodology we follow in this paper involves a gradient projected algorithm. We give a rigorous formal analysis of the proposed solution approach and establish the necessary numerical consistence properties of the resulting robust optimization algorithm.

Our contribution is devoted to constructive approximations of a wide class of the modern control systems, namely, of the general control-affine dynamic models. We consider the generic tracking control problem associated with the control-affine systems in the presence of bounded uncertainties. The conventional linearization based solution approach to the tracking problem under consideration is finally refined and extended by a novel analytic errors estimation technique. Theoretic results developed in our contribution are applied to a practically motivated example of the controlled Lotka-Volterra system.

This paper constitutes a further generalization of the numerical solution approaches to Optimal Control Problems (OCPs) of systems evolving with state suprema. We study multidimensional control systems described by differential equations with the sup-operator in the right hand sides. A specific state-observer model and the linear type feedback control design under consideration imply a resulting closed-loop system that can formally be characterized as a multidimensional Functional Differential Equation (FDE) with delays. We study OCPs associated with the obtained FDEs and establish some fundamental solution properties of this class of problems. A particular structure of the resulting dynamic optimization problem makes it possible to consider the originally given sophisticated OCP in the framework of the nonlinear separate programming in some Euclidean spaces. This fact makes it possible to apply effective and relative simple splitting type computational algorithms to the initially given sophisticated OCPs for systems evolving with state suprema.

This paper addresses the problem of robust control for a class of nonlinear dynamical systems in the continuous time domain. We deal with nonlinear models described by differential algebraic equations (DAEs) in the presence of bounded uncertainties. The full model of the control system under consideration is completed by linear sampling-type outputs. Main contribution of present manuscript is the proposal of a linear state feedback control design for this type of uncertain nonlinear system obtaining the required stability and robustness. Due to the need to the state feedback there is a need of the implementation of a Luenberguer observer. Finally, the applicability of the proposed method is illustrated by a computational example. A brief discussion on the main implementation issue is also included.

This paper proposes a new theoretic approach to a specific interaction of continuous and discrete dynamics in switched control systems known as a Zeno behaviour. We study executions of switched control systems with affine structure that admit infinitely many discrete transitions on a finite time interval. Although the real world processes do not present the corresponding behaviour, mathematical models of many engineering systems may be Zeno due to the used formal abstraction. We propose two useful approximative approaches to the Zeno dynamics, namely, an analytic technique and a variational description of this phenomenon. A generic trajectory associated with the Zeno dynamics can finally be characterized as a result of a specific projection or/and an optimization procedure applied to the original dynamic model. The obtained analytic and variational techniques provide an effective methodology for constructive approximations of the general Zeno-type behaviour. We also discuss shortly some possible applications of the proposed approximation schemes.

This article deals with applications of the newly elaborated fundamental continuity property to a class of affine switched systems and to some differential equations with discontinuous right-hand sides. We study conventional, switched and relaxed controllable dynamical models described by nonlinear ordinary differential equations that are affine in the input. The special structure of these systems makes it possible to prove the strong approximability property that can provide a useful tool for some specific robustness results. The mathematical approach based on the nonlinear and set-valued analysis, allows to consider the controllable dynamics in an abstract setting and to obtain some general theoretical results. The latter can be effectively applied to a wide classes of switched control systems and to differential inclusions with affine structure.

This paper addresses a problem of optimal control design associated with the linear networked control systems (NCSs). We study a class of the conventional networked systems in the presence of time delays and propose a hybrid LQ-based theoretical and computational approach to the above NCSs. In particular, we develop an explicit theoretical representation of the networked control processes by an adequate auxiliary hybrid systems. For the constructive feedback control design procedure we derive the necessary hybrid Riccati-formalism and propose an implementable solution procedure.

The aim of this paper is to extend the dynamic programming (DP) approach to multi-model optimal control problems (OCPs). We deal with robust optimization of multi-model control systems and are particularly interested in the Hamilton–Jacobi–Bellman (HJB) equation for the above class of problems. In this paper, we study a variant of the HJB for multi-model OCPs and examine the natural relationship between the Bellman DP techniques and the Robust Maximum Principle (MP). Moreover, we describe how to carry out the practical calculations in the context of multi-model LQ-problems and derive the associated Riccati-type equation.

In this contribution, we propose a numerical approach to optimal control processes governed by some specific switched systems. Our principal computational scheme can be characterized as an extension of the conventional proximal point method to optimal control problems (OCPs) with switched dynamical models. We study proximal based constructive approximations of the initial OCPs and establish the numerical stability (numerical consistency) of the resulting algorithm. We also discuss the practical implementability of the elaborated method and point some possible generalizations of our idea in the context of hybrid control systems.

In this paper, we deal with optimization techniques for a class of hybrid systems that comprise continuous controllable dynamics and impulses (jumps) in the state. Using the mathematical techniques of distributional derivatives and impulse differential equations, we rewrite the original hybrid control system as a system with autonomous location transitions. For the obtained auxiliary dynamical system and the corresponding optimal control problem (OCP), we apply the Lagrange approach and derive the reduced gradient formulas. Moreover, we formulate necessary optimality conditions for the above hybrid OCPs, and discuss the newly elaborated Pontryagin-type Maximum Principle for impulsive OCPs. As in the case of the conventional OCPs, the proposed first order optimization techniques provide a basis for constructive computational algorithms.

Our paper focuses on a fundamental structural property associated with a family of linear switched control systems in the presence of impulsive dynamics. We consider dynamic processes governed by piecewise linear ODEs with controlled location transitions. Using the newly developed implicit systems characterization technique, we rewrite the initially given impulsive switched system as an implicit dynamic model. This auxiliary representation of the original control system makes it possible to hide the switched phenomenon and to apply the conventional approach to the resulting implicit dynamic model. The proposed algebraic-based modeling framework follows the celebrated behavioral approach (see Polderman and Willems, 1998) and makes it possible to apply to the switched dynamics some classic techniques from the well-established time-invariant implicit systems theory. The analytic results of our paper constitute a formal theoretical extension of switched control systems methodology and can be used (as an auxiliary step) in a concrete control design procedure. The second part of our manuscript is devoted to control aspects.

This paper extends a theoretical approach to optimal control problems (OCPs) governed by a class of control systems with discontinuous right hand sides. A possible application of the framework developed in this paper is constituted by the conventional sliding mode dynamic processes. The general theory of the general constrained OCPs is finally used as an analytic basis for some conceptual numerically tractable schemes from a wide family of computational methods for OCPs. The proposed analytic method guarantees consistency of the resulting approximations related to the sophisticated initial infinite-dimensional optimization problem and can provide a fundament for some concrete implementable algorithms.

This paper deals with multiobjective optimization techniques for a class of hybrid optimal control problems in mechanical systems. We deal with a general nonlinear hybrid control systems described by boundary-value problems associated with hybrid-type Euler-Lagrange or Hamilton equations. The variational structure of the corresponding solutions makes it possible to reduce the original ”mechanical” problem to an auxiliary multiobjective programming reformulation. This approach motivates possible applications of theoretical and computational results from multiobjective optimization related to the original dynamical optimization problem. We consider first order optimality conditions for optimal control problems governed by hybrid mechanical systems and also discuss some conceptual algorithms.

Several classes of general hybrid and switched dynamic systems have been extensively studied, both in theory and practice [3,4,7,11,14,17,19,26,27,30]. In particular, driven by engineering requirements, there has been increasing interest in optimal design for hybrid control systems [3,4,7,8,13,17,23,26,27]. In this paper, we investigate some specific types of hybrid systems, namely hybrid systems of mechanical nature, and study the corresponding hybrid OCPs. The class of dynamic models to be discussed in this work concerns hybrid systems where discrete transitions are being triggered by the continuous dynamics. The control objective (control design) is to minimize a cost functional, where the control parameters are the conventional control inputs.

In this paper, we study a specific optimal control problem associated with a multiagent dynamic system. The interest is placed on minimization of the tracking error in the multiagent leader-follower model. We replace this problem by a specific hybrid optimal control problem. In particular, we consider control systems with monotonically increasing dimensions of the state vector. The change of the state dimension has the character of a jump and is modeled by an impulsive hybrid system. The paper proposes an effective computational procedure for the above optimal tracking control problem in a multiagent setting. The theoretical and numerical approaches obtained in this contribution are tested on a practically motivated example.

This paper addresses a general theoretical framework of optimal control problems (OCPs) associated with the conventional sliding mode dynamics. We deal with a class of constrained OCPs governed by nonlinear affine control systems and propose some numerically stable approximations to the sophisticated dynamical optimization problem. The above-mentioned structure of the state equations makes it also possible to consider some sensitivity properties of the optimal solutions. The mathematical approach based on the set-valued analysis allows to study the usual discontinuous sliding mode-type dynamics in the abstract setting and to obtain some general analytical results. These facts can be effectively applied to wide classes of OCPs governed by sliding mode processes and variable structure systems (VSSs).

This paper deals with an optimal design of linear network based control systems. We study a class of delayed dynamical systems in the continuous time-domain and propose an optimization approach based on the newly elaborated hybrid LQ-type techniques. Moreover, we develop an explicit connection between the given network based control processes and auxiliary hybrid models. The optimal feedback control strategy for the network based control systems is obtained as a consequence of the Riccati-formalism for the associated hybrid LQ problems. We also discuss some alternative optimization techniques and stability properties of the optimal trajectories.

This paper deals with multiobjective optimization techniques for a class of hybrid optimal control problems in mechanical systems. We deal with general nonlinear hybrid control systems described by boundary-value problems associated with hybrid-type Euler-Lagrange or Hamilton equations. The variational structure of the corresponding solutions makes it possible to reduce the original “mechanical” problem to an auxiliary multiobjective programming reformulation. This approach motivates possible applications of theoretical and computational results from multiobjective optimization related to the original dynamical optimization problem. We consider first order optimality conditions for optimal control problems governed by hybrid mechanical systems and also discuss some conceptual algorithms.

This paper deals with an optimal design of linear networked control systems. We study a class of delayed dynamical systems in the continuous time-domain and propose an optimization approach based on the newly elaborated hybrid LQ-type techniques. Moreover, we develop an explicit connection between the given networked control processes and auxiliary hybrid models. The optimal feedback control strategy for the networked systems is obtained as a consequence of the Riccati-formalism for the associated hybrid LQ problems. We also consider some alternative optimization techniques and discuss stability properties of the optimal trajectories.

This paper addresses the problem of robust control for a class of nonlinear dynamical systems in the discrete-continuous time domain. We deal with nonlinear controllable models described by ordinary differential equations in the presence of bounded uncertainties. The full model of the control system under consideration is completed by linear samplingtype outputs. The linear feedback control design proposed in this manuscript is created by application of an extended version of the conventional invariant ellipsoid method. Moreover, we also apply some specific Lyapunov-based ”descriptor techniques” from the stability theory of delayed systems. The above combination of the modified invariant ellipsoid approach and descriptor method make it possible to obtain the robustness of the designed control and to establish some well known stability properties of dynamical systems under consideration. Finally, the applicability of the proposed method is illustrated by a computational example. A brief discussion on the main implementation issue is also included.

This paper is devoted to the problem of robust design for a class of continuous-time control systems with bounded uncertainties. We study a family of nonlinearly affine dynamical system and apply a modified invariant ellipsoid technique. This makes it possible to obtain practically stable closed-loop controllable models. The design of the stabilizing feedback control strategy is based on the conventional Lyapunov-like approach to invariant sets of dynamical systems. We propose a computational scheme for a constructive treatment a feedback control law such that the region of the practical stability of the resulting system is minimized. The corresponding solution procedure contains an auxiliary LMI-constrained optimization problem. The effectiveness of the elaborated invariant ellipsoid based method is illustrated by a numerical example.

This paper is concerned with the proximal-based approachto linear and finite-difference approximations ofconstrained convex optimal control problems (OCPs). Weconsider control systems governed by ordinary differentialequations in the presence of additional terminal/state inequalitiesand propose a numerical method derived fromthe proximal point algorithm. The aim of the paper is tostudy the convergence properties of the obtained conceptualalgorithm and to show that it can be used to computeapproximate optimal controls.

This paper deals with optimization techniques for linear impulsive hybrid systems (LIHSs). We study the LQ (linear quadratic) impulsive hybrid optimal control problem (OCP) and apply the corresponding Pontryagin-type maximum principle (MP) (see). Our aim is to investigate the natural relationship between the above MP and the Bellman dynamic programming (DP) approach to the above hybrid OCP under consideration. We derive the associated Riccati-type formalism and discuss some related numerical schemes.

In this paper, we deal with optimization techniques for a class of hybrid systems that comprise continuous controllable dynamics and impulses (jumps) in the state. Using the mathematical techniques of distributional derivatives and impulse differential equations, we rewrite the original hybrid control system as a system with autonomous location transitions. For the obtained auxiliary dynamical system and the corresponding optimal control problem (OCP) we apply the Lagrange approach and derive the reduced gradient formulas. Moreover, we formulate necessary optimality conditions for the above hybrid OCPs. As in the case of a classic problem, the proposed optimization techniques also provided a basis for constructive computational algorithms.

In the present work we consider a class of nonlinear optimal control problems, which can be called "optimal control problems in mechanics". We deal with control systems whose dynamics can be described by a system of Euler-Lagrange or Hamilton equations. Using the variational structure of the solution of the corresponding boundary-value problems, we reduce the initial optimal control problem to an auxiliary problem of multiobjective programming. This technique makes it possible to apply some consistent numerical approximations of a multiobjective optimization problem to the initial optimal control problem. For solving the auxiliary problem we propose an implementable numerical algorithm.

In the present work, we consider a class of nonlinear optimal control problems, which can be called “optimal control problems in mechanics.” We deal with control systems whose dynamics can be described by a system of Euler-Lagrange or Hamilton equations. Using the variational structure of the solution of the corresponding boundary-value problems, we reduce the initial optimal control problem to an auxiliary problem of multiobjective programming. This technique makes it possible to apply some consistent numerical approximations of a multiobjective optimization problem to the initial optimal control problem. For solving the auxiliary problem, we propose an implementable numerical algorithm.

This paper deals with a quadratic optimization problem for linear impulsive hybrid systems. We study a class of LQ-type impulsive hybrid optimal control problems (OCPs) and consider the application of the hybrid Maximum Principle (MP). Our aim is to investigate the natural relationship between the Pontryagin-type MP and the Bellman Dynamic Programming (DP) approach. As next we develop the “hybrid” Riccati formalism and discuss some related computational aspects.

This paper addresses the problem of optimal design for linear networked control systems (NCSs). We deal with the delayed NCSs in the continuous time-domain and propose a computational approach to optimization problems based on the hybrid LQ-type techniques. In particular, we develop an explicit connection between the networked control processes and the corresponding hybrid systems. For the constructive feedback control design procedure we derive the necessary Riccati-formalism and propose an implementable solution algorithm.

In this paper, we investigate reachable sets for some classes of open- and closed-loop control systems governed by ordinary differential equations. We are particularly interested in convexity properties of both reachable sets and sets of trajectories. If such properties exist they can be used to estimate the reachable set under investigation. This is important in applications as, e.g., the verification of safety properties. Convexity of the set of trajectories is also important for a constructive reformulation of some constrained dynamical optimization problems in the form of a convex minimization problem.

This paper deals with a new theoretic description of some classes of the conventional sliding mode control processes. We use a variational characterization of the first-order variable structures dynamics and extend the corresponding state space formalism. A trajectory of the original systems can finally be specified as a result of a particular system optimization procedure applied to the original model. The presented variational theory of some sliding mode-type control systems is motivated by an alternative approach to the control design procedure. The mathematical tool of the sliding mode systems theory presented in our paper constitutes a formal extension of the classic Fillipov results.