Sliding and transversal motions on an inclined boundary in a periodically forced discontinuous system

In this paper, sliding and transversal motions on the boundary in the periodically driven, discontinuous dynamical system is investigated. The simple inclined straight line boundary in phase space is considered as a control law for such a dynamical system to switch. The normal vector field for a flow switching on the separation boundary is adopted to develop the analytical conditions, and the corresponding transversality conditions of a flow to the boundary are obtained. The conditions of sliding and grazing flows to the separation boundary are presented as well. Using mapping structures, periodic motions of such a discontinuous system are predicted, and the corresponding local stability and bifurcation analysis of the periodic motion are carried out. Numerical illustrations of periodic motions with and without sliding on the boundary are given. The local stability analysis cannot provide the proper prediction of the sliding and grazing motions in discontinuous dynamical systems. Therefore, the normal vector fields of periodic flows are presented, and the normal vector fields on the switching boundary points give the analytical criteria for sliding and transversality of motions.     

Asymptotic analysis of an interior optimal control problem governed by Stokes equations

This article introduces an interior optimal control problem (OCP) in a two-dimensional domain with a highly oscillatory boundary governed by the stationary Stokes equations. We consider the periodic controls in the oscillating region of the domain and use the unfolding operators to characterize the optimal controls. We establish the convergences of optimal control, state, and pressure in a suitable space to the ones of the limit system in a fixed domain.     

On motions and switchability in a periodically forced,discontinuous system with a parabolic boundary

The analytical conditions for motion switchability on the switching boundary in a periodically forced, discontinuous system are developed through the G-function of the vector fields to the switching boundary. Periodic motions in such a discontinuous dynamical system are discussed by the use of mapping structures. Two periodic motions and the analytical conditions are presented for illustration. Further investigation should be carried out for a better understanding of the vanishing and stability of regular and chaotic motions.     

Computing eigenvalues of Sturm-Liouville problems via optimal control theory

In this paper, we shall address three problems arising in the computation of eigenvalues of Sturm-Liouville boundary value problems. We first consider a well-posed Sturm-Liouville problem with discrete and distinct spectrum. For this problem, we shall show that the eigenvalues can be computed by solving for the zeros of the boundary condition at the terminal point as a function of the eigenvalue. In the second problem, we shall consider the case where some coefficients and parameters in the differential equation are continuously adjustable. For this, the eigenvalues can be optimized with respect to these adjustable coefficients and parameters by reformulating the problem as a combined optimal control and optimal parameter selection problem. Subsequently, these optimized eigenvalues can be computed by using an existing optimal control software, MISER. The last problem extends the first to nonstandard boundary conditions such as periodic or interrelated boundary conditions. To illustrate the efficiency and the versatility of the proposed methods, several non-trivial numerical examples are included.     

Near-optimal controls of differential systems with switching and random jumps subject to fast switching and wideband noise perturbation

This work develops near-optimal controls for systems given by differential equations with wideband noise and random switching. The random switching is modeled by a continuous-time, time-inhomogeneous Markov chain. Under broad conditions, it is shown that there is an associated limit problem, which is a switching jump diffusion. Using near-optimal controls of the limit system, we then build controls for the original systems. It is shown that such constructed controls are nearly optimal.     

Flow switchability and periodic motions in a periodically forced,discontinuous dynamical system

This paper presents the switchability of a flow from one domain into another one in the periodically forced, discontinuous dynamical system. The inclined line boundary in phase space is used for the dynamical system to switch. The normal vector field product for flow switching on the separation boundary is introduced. The passability condition of a flow to the separation boundary is achieved through such a normal vector field product, and the sliding and grazing conditions to the separation boundary are presented as well. Using mapping structures, periodic motions in such a discontinuous system are predicted analytically, and the corresponding local stability and bifurcation analysis are carried out. With the analytical conditions of grazing and sliding motions, the parameter maps of specific motions are developed. Illustrations of periodic and chaotic motions are given, and the normal vector fields are presented to show the analytical criteria. This investigation may help one better understand the sliding mode control. The methodology presented in this paper can be applied to discontinuous, nonlinear systems.     

On second-order necessary optimality conditions in the classical sense for systems with nonlocal conditions

We consider an optimal control problem in which the state of a system is described by control systems of ordinary differential equations with two-point boundary conditions. The admissible controls are chosen in the class of bounded measurable functions. We derive a formula for the increment of a second-order functional. On the basis of control variations, we obtain necessary optimality conditions for singular controls in the classical sense.     

Numerical solution of minimax optimal control problems by multiple shooting technique

This paper presents the application of the multiple shooting technique to minimax optimal control problems (optimal control problems with Chebyshev performance index). A standard transformation is used to convert the minimax problem into an equivalent optimal control problem with state variable inequality constraints. Using this technique, the highly developed theory on the necessary conditions for state-restricted optimal control problems can be applied advantageously. It is shown that, in general, these necessary conditions lead to a boundary-value problem with switching conditions, which can be treated numerically by a special version of the multiple shooting algorithm. The method is tested on the problem of the optimal heating and cooling of a house. This application shows some typical difficulties arising with minimax optimal control problems, i.e., the estimation of the switching structure which is dependent on the parameters of the problem. This difficulty can be overcome by a careful application of a continuity method. Numerical solutions for the example are presented which demonstrate the efficiency of the method proposed.     

Solving elliptic control problems with interior point and SQP methods: control and state constraints

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. Both boundary control and distributed control problems are considered with boundary conditions of Dirichlet or Neumann type. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. Necessary conditions of optimality are discussed both for the continuous and the discretized control problem. It is shown that the recently developed interior point method LOQO of [35] is capable of solving these problems even for high discretizations. Four numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang–bang controls.     

The Optimal Control of a Train

We consider the problem of determining an optimal driving strategy in a train control problem with a generalised equation of motion. We assume that the journey must be completed within a given time and seek a strategy that minimises fuel consumption. On the one hand we consider the case where continuous control can be used and on the other hand we consider the case where only discrete control is available. We pay particular attention to a unified development of the two cases. For the continuous control problem we use the Pontryagin principle to find necessary conditions on an optimal strategy and show that these conditions yield key equations that determine the optimal switching points. In the discrete control problem, which is the typical situation with diesel-electric locomotives, we show that for each fixed control sequence the cost of fuel can be minimised by finding the optimal switching times. The corresponding strategies are called strategies of optimal type and in this case we use the Kuhn–Tucker equations to find key equations that determine the optimal switching times. We note that the strategies of optimal type can be used to approximate as closely as we please the optimal strategy obtained using continuous control and we present two new derivations of the key equations. We illustrate our general remarks by reference to a typical train control problem.     

Optimal Bang-Bang Controls for a Two-Compartment Model in Cancer Chemotherapy

A class of mathematical models for cancer chemotherapy which have been described in the literature take the form of an optimal control problem over a finite horizon with control constraints and dynamics given by a bilinear system. In this paper, we analyze a two-dimensional model in which the cell cycle is broken into two compartments. The cytostatic agent used as control to kill the cancer cells is active only in the second compartment where cell division occurs and the cumulative effect of the drug is used to model the negative effect of the treatment on healthy cells. It is shown that singular controls are not optimal for this model and the optimality properties of bang-bang controls are established. Specifically, transversality conditions at the switching surfaces are derived. In a nondegenerate setting, these conditions guarantee the local optimality of the flow if satisfied, while trajectories will be no longer optimal if they are violated.     

Optimal control of impulsive systems with nonlocal boundary conditions

In this paper we study an optimal control problem, where states of a control system are described by impulsive differential equations with nonlocal boundary conditions. With the help of the contraction principle we prove the existence and uniqueness of a solution to the corresponding boundary value problem with fixed admissible controls. We calculate the first and second variation of the functional. Using the variation of controls, we establish various necessary optimality conditions of the second order.     

Quadratic control for linear periodic systems

We propose a new quadratic control problem for linear periodic systems which can be finite or infinite dimensional. We consider both deterministic and stochastic cases. It is a generalization of average cost criterion, which is usually considered for time-invariant systems. We give sufficient conditions for the existence of periodic solutions.Under stabilizability and detectability conditions we show that the optimal control is given by a periodic feedback which involves the periodic solution of a Riccati equation. The optimal closed-loop system has a unique periodic solution which is globally exponentially asymptotically stable. In the stochastic case we also consider the quadratic problem under partial observation. Under two sets of stabilizability and detectability conditions we obtain the separation principle. The filter equation is not periodic, but we show that it can be effectively replaced by a periodic filter. The theory is illustrated by simple examples.This work was done while this author was a visiting professor at the Scuola Normale Superiore, Pisa.     

Piecewise-Linear Pathways to the Optimal Solution Set in Linear Programming

An optimal control problem with four linear controls describing a sophisticated concern model is investigated. The numerical solution of this problem by combination of a direct collocation and an indirect multiple shooting method is presented and discussed. The approximation provided by the direct method is used to estimate the switching structure caused by the four controls occurring linearly. The optimal controls have bang-bang subarcs as well as constrained and singular subarcs. The derivation of necessary conditions from optimal control theory is aimed at the subsequent application of an indirect multiple shooting method but is also interesting from a mathematical point of view. Due to the linear occurrence of the controls, the minimum principle leads to a linear programming problem. Therefore, the Karush–Kuhn–Tucker conditions can be used for an optimality check of the solution obtained by the indirect method.     

Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints: Part 1. Boundary Control

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. In a first part we consider boundary control problems with either Dirichlet or Neumann conditions. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. It is shown that a recently developed interior point method is able to solve these problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang and singular controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints.     

Abort landing in the presence of windshear as a minimax optimal control problem,part 2: Multiple shooting and homotopy

In Part 1 of the paper (Ref. 2), we have shown that the necessary conditions for the optimal control problem of the abort landing of a passenger aircraft in the presence of windshear result in a multipoint boundary-value problem. This boundary-value problem is especially well suited for numerical treatment by the multiple shooting method. Since this method is basically a Newton iteration, initial guesses of all variables are needed and assumptions about the switching structure have to be made. These are big obstacles, but both can be overcome by a so-called homotopy strategy where the problem is imbedded into a one-parameter family of subproblems in such a way that (at least) the first problem is simple to solve. The solution data to the first problem may serve as an initial guess for the next problem, thus resulting in a whole chain of problems. This process is to be continued until the objective problem is reached.Techniques are presented here on how to handle the various changes of the switching structure during the homotopy run. The windshear problem, of great interest for safety in aviation, also serves as an excellent benchmark problem: Nearly all features that can arise in optimal control appear when solving this problem. For example, the candidate for an optimal trajectory of the minimax optimal control problem shows subarcs with both bang-bang and singular control functions, boundary arcs and touch points of two state constraints, one being of first order and the other being of third order, etc. Therefore, the results of this paper may also serve as some sort of user's guide for the solution of complicated real-life optimal control problems by multiple shooting.The candidate found for an optimal trajectory is discussed and compared with an approximate solution already known (Refs. 3–4). Besides the known necessary conditions, additional sharp necessary conditions based on sign conditions of certain multipliers are also checked. This is not possible when using direct methods.An extended abstract of this paper was presented at the 8th IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Paris, France, 1989 (see Ref. 1).This paper is dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.     

Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator

The discontinuous dynamics of a non-linear, friction-induced, periodically forced oscillator is studied. The analytical conditions for motion switchability at the velocity boundary in such a nonlinear oscillator are developed to understand the motion switching mechanism. Using such analytical conditions of motion switching, numerical predictions of the switching scenarios varying with excitation frequency and amplitude are carried out, and the parameter maps for specific periodic motions are presented. Chaotic and periodic motions are illustrated through phase planes and switching sections for a better understanding of motion mechanism of the nonlinear friction oscillator. The periodic motions with switching in such a nonlinear, frictional oscillator cannot be obtained from the traditional analysis (e.g., perturbation and harmonic balance method).     

Optimal boundary displacement control of string vibrations with a nonlocal oddness condition of the first kind

We consider the problem of optimal boundary control by the displacement at left endpoint of a string in the case of a nonlocal oddness boundary condition of the first kind. We obtain a necessary and sufficient condition for the problem controllability under arbitrary initial and terminal conditions and construct a closed analytical form of the control itself under these conditions. In addition, we consider the problem of optimal boundary control by the displacement at one endpoint of the string for a given displacement mode at the other endpoint.