The existence of optimal control on the basis of Weierstrass’s theorem

The problem of existence of an optimal control is solved on the basis of Weierstrass’s classical theorem if the set of admissible controls belongs to the class of piecewise continuous functions. In the process of describing admissible controls, the main assumption is that the number of switchings (points of discontinuity) is uniformly bounded and not just finite, as in the main problem of optimal control theory. On the one hand, this assumption does not restrict the spectrum of optimal control applications. On the other hand, it fits the Weierstrass’s theorem owing to the convenience in characterizing the sequential compactness. The formulation of Weierstrass’s theorem, which asserts the existence of continuous function extrema on sequentially compact sets, is customary, and its proof complies with the traditional scheme, whereas the concepts (convergent sequences and some others) are adapted to the peculiarity of optimal problems.     

Regimes of more and more frequent switchings in the optimal control problem of oscillations of <Emphasis Type="Italic">n</Emphasis> oscillators

This paper considers the control system of n oscillators executing forced oscillations under the action of a scalar-valued control force common for all oscillators whose module is bounded. The author proves the existence of an optimal singular regime and the assertion that the optimal control has at least countably many switchings that accumulate to a conjunction point of the singular and nonsingular parts of the trajectory. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 19, Optimal Control, 2006.     

The Pontryagin derivative in optimal control

A basic feature of Pontryagin’s maximum principle is its native Hamiltonian format, inherent in the principle regardless of any regularity conditions imposed on the optimal problem under consideration. It canonically assigns to the problem a family of Hamiltonian systems, indexed with the control parameter, and complements the family with the maximum condition, which makes it possible to solve the initial value problem for the system by “dynamically” eliminating the parameter as we proceed along the trajectory, thus providing extremals of the problem. Much has been said about the maximum condition since its discovery in 1956, and all achievements in the field were mainly credited to it, whereas the Hamiltonian format of the maximum principle has always been taken for granted and never been discussed seriously. Meanwhile, the very possibility of formulating the maximum principle is intimately connected with its native Hamiltonian format and with the parametrization of the problem with the control parameter. Both these starting steps were made by L.S. Pontryagin in 1955 from scratch, in fact, out of nothing, and eventually led to the discovery of the maximum principle. Since the present volume is dedicated to the centenary of the birth of Lev Semenovich Pontryagin, I decided to return to this now semi-historical topic and give a short exposition of the Hamiltonian format of the maximum principle.     

Optimal singular and chattering modes in the problem of controlling the vibrations of a string with clamped ends

The problem of minimizing the root mean square deviation of a uniform string with clamped ends from an equilibrium position is investigated. It is assumed that the initial conditions are specified and the ends of the string are clamped. The Fourier method is used, which enables the control problem with a partial differential equation to be reduced to a control problem with a denumerable system of ordinary differential equations. For the optimal control problem in the l₂ space obtained, it is proved that the optimal synthesis contains singular trajectories and chattering trajectories. For the initial problem of the optimal control of the vibrations of a string it is also proved that there is a unique solution for which the optimal control has a denumerable number of switchings in a finite time interval.     

Relaxation methods for mixed-integer optimal control of partial differential equations

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments.     

Optimal synthesis in a two-dimensional problem with asymmetric constraints on the control

A two-dimensional optimal control problem is considered on the assumption that the terminal time of the process is not fixed and the integral objective functional depends on a parameter. Asymmetric constraints are imposed on the control parameter. Two cases are considered: constraints of the same sign and constraints of different signs. In the case of constraints of different signs, if the parameters of the problem satisfy certain relations, one obtains chattering control, alternating with a control with two switchings and a first-order singular are when these relations are violated. In the case of sign-definite control the controllability domain is part of the plane bounded by two semiparabolas. Three types of control law are then possible, in two of which the system will hit the boundary of the controllability domain and move along it, while the third features a first-order singular are. As the parameter of the problem is varied, the phase portrait undergoes evolution and one of these three types is interchanged with another. The optimality of these control laws is rigorously established using a dynamic programming method.     

Optimum maneuvers of a skip vehicle with bounded lift constraints

This paper presents the analytical solutions of the problem of optimum maneuvering of a glide vehicle flying in the hypervelocity regime. The investigation is based on the approximation of Allen and Eggers; namely, that along the fundamental part of a reentry or ascent trajectory, the aerodynamic forces greatly exceed the components of the gravitational force in the directions tangent and normal to the flight path.The problem consists of finding an optimal control law for the lift such that the final velocity or the final altitude is maximized. This problem can be viewed as bringing the vehicle to the best condition for interception, penetration, or making an evasive maneuver.If the range is free, the optimal lift control is obtained in closed form. If the lift control is bounded, then bounded control is optimal whenever it is reached. The switching sequences for different cases are discussed, and it is shown that there are at most two switchings. Bounded lift control is always at the ends of the optimal trajectory; for the case of two switchings, the optimal trajectory has an inflection point.The authors wish to thank the National Aeronautics and Space Administration for the Grant No. NGR-06-003-033 under which this work was carried out.     

Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints

We consider the generalized Nash equilibrium problem (GNEP), in which each player’s strategy set may depend on the rivals’ strategies through shared constraints. A practical approach to solving this problem that has received increasing attention lately entails solving a related variational inequality (VI). From the viewpoint of game theory, it is important to find as many GNEs as possible, if not all of them. We propose two types of parametrized VIs related to the GNEP, one price-directed and the other resource-directed. We show that these parametrized VIs inherit the monotonicity properties of the original VI and, under mild constraint qualifications, their solutions yield all GNEs. We propose strategies to sample in the parameter spaces and show, through numerical experiments on benchmark examples, that the GNEs found by the parametrized VI approaches are widely distributed over the GNE set.     

Dinkelbach Approach to Solving a Class of Fractional Optimal Control Problems

We consider optimal control problems with functional given by the ratio of two integrals (fractional optimal control problems). In particular, we focus on a special case with affine integrands and linear dynamics with respect to state and control. Since the standard optimal control theory cannot be used directly to solve a problem of this kind, we apply Dinkelbach’s approach to linearize it. Indeed, the fractional optimal control problem can be transformed into an equivalent monoparametric family {P_q} of linear optimal control problems. The special structure of the class of problems considered allows solving the fractional problem either explicitly or requiring straightforward classical numerical techniques to solve a single equation. An application to advertising efficiency maximization is presented. This work was partially supported by the Università Ca’ Foscari, Venezia, Italy, the MIUR (PRIN cofinancing 2005), the Council for Grants (under RF President) and State Aid to Fundamental Science Schools (Grant NSh-4113.2008.6). We thank Angelo Miele, Panos Pardalos and the anonymous referees for comments and suggestions.     

Inverse optimal control and construction of control Lyapunov functions

In this paper, the construction of CLFs for nonlinear systems and a new inverse optimal control law are presented. The construction of a CLF for an affine nonlinear system is reduced to the construction of a CLF for a simpler system, and a new L _g V type control law with respect to a CLF is provided. This control law is a generalization of Sontag’s formula and contains a design parameter. Tuning this parameter gives many suboptimal solutions for the optimization problem. Also, the gain margin and sector margin of the control law are calculated. Examples are provided to illustrate the main theoretical results of the paper. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

Periodic switching strategies for an isoperimetric control problem with application to nonlinear chemical reactions

This paper deals with an isoperimetric optimal control problem for nonlinear control-affine systems with periodic boundary conditions. As it was shown previously, the candidates for optimal controls for this problem can be obtained within the class of bang-bang input functions. We consider a parametrization of these inputs in terms of switching times. The control-affine system under consideration is transformed into a driftless system by assuming that the controls possess properties of a partition of unity. Then the problem of constructing periodic trajectories is studied analytically by applying the Fliess series expansion over a small time horizon. We propose analytical results concerning the relation between the boundary conditions and switching parameters for an arbitrary number of switchings. These analytical results are applied to a mathematical model of non-isothermal chemical reactions. It is shown that the proposed control strategies can be exploited to improve the reaction performance in comparison to the steady-state operation mode.     

Optimal design and control of queues

We have divided this review into two parts. The first part is concerned with the optimal design of queueing systems and the second part deals with the optimal control of queueing systems. The second part, which has the lion’s share of the review since it has received the most attention, focuses mainly on the modelling aspects of the problem and describes the different kinds of threshold (control) policy models available in the literature. To limit the scope of this survey, we decided to limit ourselves to research on papers dealing with the three policies (N, T, and D), where a cost function is designed specifically and optimal thresholds that yield minimum cost are sought.     

Tracking a reference solution of a control system of phase field equations

We consider the problem of tracking a reference solution of a dynamical system described by a pair of distributed differential equations, the phase field equations. To solve this problem, we propose an algorithm based on Yu.S. Osipov’s theory of dynamic inversion and on N.N. Krasovskii’s extremal shift method developed in the theory of positional differential games.     

Control of interconnected nonlinear delay differential equations inW 2 (1)

Our main interest in this paper is the resolution of the problem of controllability of interconnected nonlinear delay systems in function space, from which hopefully the existence of an optimal control law can be deduced later. We insist that each subsystem be controlled by its own variables while taking into account the interacting effects. This is the recent basic insight of [13] on ordinary differential systems. Controllability is deduced for the composite system from the assumption of controllability of each free subsystem and a growth condition of the interconnecting structure. Conditions for a free system’s controllability are given. One application is presented. The insight it provides for the growth of global economy has important policy implications.     

广义分布参数系统的状态反馈稳定性问题(1)

关于系统的状态反馈稳定性问题的研究一直是现代控制理论研究的重要问题之一.广义分布参数系统是比分布参数系统更广的一类系统,在研究复合材料热导体中的温度分布等问题时会出现这样的系统.本文讨论了H ilbert空间中一阶广义分布参数系统的状态反馈稳定性问题.应用泛函分析及线性算子半群理论的方法给出了使闭环广义分布参数系统渐进稳定的充要条件,充分条件及状态反馈的构造性表达式.这对研究广义分布参数系统的状态反馈稳定性问题具有重要的理论价值.     

Optimal chattering modes in the problem of the control of a Timoshenko beam

The linear problem of the control in a plane of the motion of a Timoshenko beam, one end of which is clamped to a rotating disc is considered. The angular acceleration of the disc serves as the control. It is proved that, in the problem of the quenching of the first mode, the optimal control has an infinite number of switchings in a finite time interval (a chattering control). The construction of a suboptimal control with a finite number of switchings is described.     

OPTIMAL HARVESTING AND SPATIAL PATTERNS IN A SEMIARID VEGETATION SYSTEM

We consider an infinite time horizon spatially distributed optimal harvesting problem for a vegetation and soil water reaction diffusion system, with rainfall as the main external parameter. By Pontryagin's maximum principle, we derive the associated four‐component canonical system (CS), and numerically analyze this and hence the optimal control problem in two steps. First, we numerically compute a rather rich bifurcation structure of flat (spatially homogeneous) canonical steady states and patterned canonical steady states (FCSS and PCSS, respectively), in 1D and 2D. Then, we compute time‐dependent solutions of the CS that connect to some FCSS or PCSS. The method is efficient in dealing with nonunique canonical steady states, and thus also with multiple local maxima of the objective function. It turns out that over wide parameter regimes the FCSS, i.e., spatially uniform harvesting, are not optimal. Instead, controlling the system to a PCSS yields a higher profit. Moreover, compared to (a simple model of) private optimization, the social control gives a higher yield, and vegetation survives for much lower rainfall. In addition, the computation of the optimal (social) control gives an optimal tax to incorporate into the private optimization.     

Reduced order controllers for distributed parameter systems: LQG balanced truncation and an adaptive approach

In this paper, two methods are reviewed and compared for designing reduced order controllers for distributed parameter systems. The first involves a reduction method known as LQG balanced truncation followed by MinMax control design and relies on the theory and properties of the distributed parameter system. The second is a neural network based adaptive output feedback synthesis approach, designed for the large scale discretized system and depends upon the relative degree of the regulated outputs. Both methods are applied to a problem concerning control of vibrations in a nonlinear structure with a bounded disturbance.     

k-out-of-n-system with repair:T-policy

We consider a k-out-of-n system with repair underT-policy. Life time of each component is exponentially distributed with parameter λ. Server is called to the system after the elapse ofT time units since his departure after completion of repair of all failed units in the previous cycle or until accumulation ofn — k failed units, whichever occurs first. Service time is assumed to be exponential with rateμ.T is also exponentially distributed with parameter α. System state probabilities in finite time and long run are derived for (i) cold (ii) warm (iii) hot systems. Several characteristics of these systems are obtained. A control problem is also investigated and numerical illustrations are provided. It is proved that the expected profit to the system is concave in α and hence global maximum exists.