Optimal control of impulsive switched systems with minimum subsystem durations

This paper presents a new computational approach for solving optimal control problems governed by impulsive switched systems. Such systems consist of multiple subsystems operating in succession, with possible instantaneous state jumps occurring when the system switches from one subsystem to another. The control variables are the subsystem durations and a set of system parameters influencing the state jumps. In contrast with most other papers on the control of impulsive switched systems, we do not require every potential subsystem to be active during the time horizon (it may be optimal to delete certain subsystems, especially when the optimal number of switches is unknown). However, any active subsystem must be active for a minimum non-negligible duration of time. This restriction leads to a disjoint feasible region for the subsystem durations. The problem of choosing the subsystem durations and the system parameters to minimize a given cost function is a non-standard optimal control problem that cannot be solved using conventional techniques. By combining a time-scaling transformation and an exact penalty method, we develop a computational algorithm for solving this problem. We then demonstrate the effectiveness of this algorithm by considering a numerical example on the optimization of shrimp harvesting operations.     

Applications of numerical optimal control to nonlinear hybrid systems

This paper develops a technique for numerically solving hybrid optimal control problems. The theoretical foundation of the approach is a recently developed methodology by S.C. Bengea and R.A. DeCarlo [Optimal control of switching systems, Automatica. A Journal of IFAC 41 (1) (2005) 11–27] for solving switched optimal control problems through embedding. The methodology is extended to incorporate hybrid behavior stemming from autonomous (uncontrolled) switches that results in plant equations with piecewise smooth vector fields. We demonstrate that when the system has no memory, the embedding technique can be used to reduce the hybrid optimal control problem for such systems to the traditional one. In particular, we show that the solution methodology does not require mixed integer programming (MIP) methods, but rather can utilize traditional nonlinear programming techniques such as sequential quadratic programming (SQP). By dramatically reducing the computational complexity over existing approaches, the proposed techniques make optimal control highly appealing for hybrid systems. This appeal is concretely demonstrated in an exhaustive application to a unicycle model that contains both autonomous and controlled switches; optimal and model predictive control solutions are given for two types of models using both a minimum energy and minimum time performance index. Controller performance is evaluated in the presence of a step frictional disturbance and parameter uncertainties which demonstrates the robustness of the controllers.     

Error Estimates for the Numerical Approximation of a Semilinear Elliptic Control Problem

We study the numerical approximation of distributed nonlinear optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Our main result are error estimates for optimal controls in the maximum norm. Characterization results are stated for optimal and discretized optimal control. Moreover, the uniform convergence of discretized controls to optimal controls is proven under natural assumptions.     

Analysis of the Optimal Relaxed Control to an Optimal Control Problem

Relaxed controls are widely used to analyze the existence of optimal controls in the literature. Though there are many optimal control problems admitting no optimal control, rare examples were shown. This paper will solve a particular optimal control problem by analyzing the optimal relaxed controls, showing the ideas we used to study such kind of problems. This work was supported by NSFC (No. 10671040), FANEDD (No. 200522) and NCET (No. 06-0359).     

Time optimal sampled-data controls for the heat equation

In this paper, we first design a time optimal control problem for the heat equation with sampled-data controls, and then use it to approximate a time optimal control problem for the heat equation with distributed controls.The study of such a time optimal sampled-data control problem is not easy, because it may have infinitely many optimal controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem, and obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the time optimal sampled-data control problem and the time optimal distributed control problem, in terms of the sampling period, but we also prove that such estimates are optimal in some sense.     

Optimal control of a dynamical system representing a gantry crane

Problems arising in the optimal control of gantry crane instaliations are considered. Continuous controls to minimize a control squared objective function are obtained. The amplitude of in-plane oscillations of the suspended mass is assumed small. The optimal controls are sufficiently simple for practical realization.     

On linear nonstationary control processes

For a linear nonstationary optimal control problem, the number of switches in a time-optimal piecewise constant control is estimated above in the case where the control set U is a convex polytope and a genericity condition holds at all points of the time interval under consideration.     

Controllability,extremality, and abnormality in nonsmooth optimal control

We derive sufficient conditions for controllability and necessary conditions for minimum in nonsmooth optimal control problems defined by differential or functional-integral equations with isoperimetric and unilateral restrictions. We consider the cases when the controls are relaxed or chosen fromabundant sets of original (ordinary) controls (which include most, or all, of the control sets studied in the literature). We prove that, if there exist optimal strictly original controls (that is, controls that are optimal in an abundant set but not among relaxed controls), then the problem admits abnormal extremals. We also study the abnormality of the optimal strictly original controls themselves.     

Some remarks on the numerical solution of bang-bang type optimal control problems

This paper is concerned with the numerical solution of optimal control problems for which each optimal control is bang-bang. Especially, the results apply to parabolic boundary control Problems. Starting from a sequence of feasible solutions converging to an optimal control u, a sequence of bang-bang controls converging to u is constructed. Bang-bang approximations of u are desirable for certain numerical reasons. Sequences of arbitrary feasible controls converging to u may be obtained by discretization or by a descent method. Numerical examples are also given.     

Approximation of relaxed nonlinear parabolic optimal control problems

We consider a relaxed optimal control problem for systems defined by nonlinear parabolic partial differential equations with distributed control. The problem is completely discretized by using a finite-element approximation scheme with piecewise linear states and piecewise constant controls. Existence of optimal controls and necessary conditions for optimality are derived for both the continuous and the discrete problem. We then prove that accumulation points of sequences of discrete optimal [resp. extremal] controls are optimal [resp. extremal] for the continuous problem.     

Near-Optimal Controls of a Class of Volterra Integral Systems

In a recent paper by Zhou (Ref. 1), the concept of near-optimal controls was introduced for a class of optimal control problems involving ordinary differential equations. Necessary and sufficient conditions for near-optimal controls were derived. This paper extends the results obtained by Zhou to a class of optimal control problems involving Volterra integral equations. The results are applied to study near-optimal controls obtained by the control parametrization method.     

Control Localized on Thin Structures for the Linearized Boussinesq System

We study optimal control problems for the linearized Boussinesq system when the control is supported on a submanifold of the boundary of the domain. This type of problem belongs to the class of optimal control problems with measures as controls, which has been studied recently by several authors. We are mainly interested in the optimality conditions for such problems. It is known that the differentiability properties needed to obtain the optimality conditions are more demanding, in terms of regularity of the data, than what is needed to prove the existence of optimal controls. Here we are able to derive the optimality conditions by taking advantage of the particular structure of the controls.     

Third-order nilpotency, nice reachability and asymptotic stability

We consider an affine control system whose vector fields span a third-order nilpotent Lie algebra. We show that the reachable set at time T using measurable controls is equivalent to the reachable set at time T using piecewise-constant controls with no more than four switches. The bound on the number of switches is uniform over any final time T. As a corollary, we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching. This provides a partial solution to an open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton Univ. Press, 2004, pp. 203-207].     

Asymptotic analysis of Neumann periodic optimal boundary control problem

An optimal boundary control problem in a domain with oscillating boundary has been investigated in this paper. The controls are acting periodically on the oscillating boundary. The controls are applied with suitable scaling parameters. One of the major contribution is the representation of the optimal control using the unfolding operator. We then study the limiting analysis (homogenization) and obtain two limit problems according to the scaling parameters. Another notable observation is that the limit optimal control problem has three controls, namely, a distributed control, a boundary control, and an interface control. Copyright © 2016 John Wiley & Sons, Ltd.     

Random relaxed controls and partially observed stochastic systems

We introduce a class of generalized controls called random relaxed controls, and show that under quite general conditions, a partially observed, controlled diffusion will have an optimal random relaxed control whose cost equals the infimum over the costs of all ordinary controls. We also show that the optimal admissible control can be approximated arbitrarily well by very simple, ordinary controls. The proofs are based on a close analysis of the standard parts of nonstandard controls.     

Switching control and time-delay identification

The unknown time delay makes the control design a difficult task. When the lower and upper bounds of an unknown time delay of dynamical systems are specified, one can design a supervisory control that switches among a set of controls designed for the sampled time delays in the given range so that the closed-loop system is stable and the control performance is maintained at a desirable level. In this paper, we propose to design a supervisory control to stabilize the system first. After the supervisory control converges, we start an algorithm to identify the unknown time delay, either on-line or off-line, with the known control being implemented. Examples are shown to demonstrate the stabilization and identification for linear time invariant and periodic systems with a single control time delay.     

Error Estimates for the Approximation of a Class of Optimal Control Systems Governed by Linear PDEs

This paper deals with a class of optimal control problems in which the system is governed by a linear partial differential equation and the control is distributed and with constraints. The problem is posed in the framework of the theory of optimal control of systems. A numerical method is proposed to approximate the optimal control. In this method, the state space as well as the convex set of admissible controls are discretized. An abstract error estimate for the optimal control problem is obtained that depends on both the approximation of the state equation and the space of controls. This theoretical result is illustrated by some numerical examples from the literature.     

Existence of Optimal Controls in the Absence of Cesari-Type Conditions forSemilinear Elliptic and Parabolic Systems

Some new results on the existence of optimal controls are established for control systems governed by semilinear elliptic or parabolic equations. No Cesari type conditions are assumed. By proving existence theorems and analyzing the Pontryagin maximum principle for optimal relaxed state-control pairs for the corresponding relaxed problems, existence theorems of classical optimal pairs for the original problem are established. To treat the case of a noncompact control set, relaxed controls defined by finitely additive measures are used.     

Deterministic and Stochastic Control of Navier—Stokes Equation with Linear, Monotone, and Hyperviscosities

This paper deals with the optimal control of space—time statistical behavior of turbulent fields. We provide a unified treatment of optimal control problems for the deterministic and stochastic Navier—Stokes equation with linear and nonlinear constitutive relations. Tonelli type ordinary controls as well as Young type chattering controls are analyzed. For the deterministic case with monotone viscosity we use the Minty—Browder technique to prove the existence of optimal controls. For the stochastic case with monotone viscosity, we combine the Minty—Browder technique with the martingale problem formulation of Stroock and Varadhan to establish existence of optimal controls. The deterministic models given in this paper also cover some simple eddy viscosity type turbulence closure models. Accepted 7 June 1999     

Convergence properties of optimal controls of pseudo-parabolic problems

Approximation properties of pseudo-parabolic optimal controls to the parabolic optimal control are considered for two concrete problems.