Time-inconsistent stochastic linear-quadratic control problem with indefinite control weight costs

A time-inconsistent linear-quadratic optimal control problem for stochastic differential equations is studied. We introduce conditions where the control cost weighting matrix is possibly singular. Under such conditions, we obtain a family of closed-loop equilibrium strategies via multi-person differential games. This result extends Yong’s work(2017) in the case of stochastic differential equations, where a unique closed-loop equilibrium strategy can be derived under standard conditions(namely, t...     

Some aspects of reachability for parabolic boundary control problems with control constraints

A class of one-dimensional parabolic optimal boundary control problems is considered. The discussion includes Neumann, Robin, and Dirichlet boundary conditions. The reachability of a given target state in final time is discussed under box constraints on the control. As a mathematical tool, related exponential moment problems are investigated. Moreover, based on a detailed study of the adjoint state, a technique is presented to find the location and the number of the switching points of optimal bang-bang controls. Numerical examples illustrate this procedure.     

Optimal ℋ∞-state feedback control for continuous-time linear systemsfeedback control for continuous-time linear systems

This paper proposes a convex programming method to achieve optimal -state feedback control for continuous-time linear systems. State space conditions, formulated in an appropriate parameter space, define a convex set containing all the stabilizing control gains that guarantee an upper bound on the -norm of the closed-loop transfer function. An optimization problem is then proposed, in order to minimize this upper bound over the previous convex set, furnishing the optimal -control gain as its optimal solution. A limiting bound for the optimum -norm can easily be calculated, and the proposed method will achieve minimum attenuation whenever a feasible state feedback controller exists. Generalizations to decentralized and output feedback control are also investigated. Numerical examples illustrate the theory.This research has been supported in part by grants from Fundação de Amparo à Pesquisa do Estado de São Paulo—FAPESP and Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq—Brazil. The authors are grateful to the anonymous referees for their useful comments on this paper.     

Γ-convergence and optimal control problems

In this paper, we give some applications ofG-convergence and -convergence to the study of the asymptotic limits of optimal control problems. More precisely, given a sequence (P _h) of optimal control problems and a control problem (P), we determine some general conditions, involvingG-convergence and -convergence, under which the sequence of the optimal pairs of the problems (P _h) converges to the optimal pair of problem (P).The authors wish to thank Professor E. De Giorgi for many stimulating discussions.     

An optimal control problem inL ∞1

We consider an optimal control problem inL , where the cost functional has a penalty term which involves the structure of the control law. This type of penalization allows us to restrict our attention only to piecewise constant controls, with assigned switching times, i.e., the controls belong to finite-dimensional control spaces. This fact and our assumptions on the dynamics give as a consequence the compactness of the minimizing sequences inC([0, 1], ) ×L ([0, 1], ). The existence of a minimum of the cost functional is then obtained by a direct method. This result allows us to avoid the usual convexity assumption on the cost functionalC and on the multivalued vector field associated to the dynamics when we have to consider the controls in all ofL .This research was partially supported by Research Project M.P.I. 40%, Teoria del Controllo dei Sistemi Dinamici.     

Saab 500 — A numerical control system

The content of the first part is a short analysis of the concept Numerical Control and a discussion of associated procedures. The hardware part assigned to the controlled process is constructed as a control calculator. After a discussion of the amount of information required to feed the control calculator, further considerations have lead to the ideas behind the Saab 500 numerical control system.The Saab 500 system is presented in its main parts.     

Sampled-data control of Lur’e systems

The main purpose of this paper is to fill a gap in the literature concerning the problem of designing a sampled-data control law for continuous-time Lur’e systems. The goal is to design a state feedback sampled-data control for this class of nonlinear systems preserving global asymptotic stability and minimizing a guaranteed quadratic cost. The main challenge towards the solution of the proposed problem is to handle this class of nonlinear system in order to propose less conservative design conditions expressed through differential linear matrix inequalities - (DLMIs). Bellman’s Principle of Optimality applied together with the Popov–Lyapunov function that emerges from the celebrated Popov Stability Criterion is the key issue to obtain the reported results. Two examples are solved for illustration.     

Fractals control in particle’s velocity

Julia set, a fractal set of the literature of nonlinear physics, has significance for the engineering applications. For example, the fractal structure characteristics of the generalized M–J set could visually reflect the change rule of particle’s velocity. According to the real world requirement, the system need show various particle’s velocity in some cases. Thus, the control of the nonlinear behavior, i.e., Julia set, has attracted broad attention. In this work, an auxiliary feedback control is introduced to effectively control the Julia set that visually reflects the change rule of particle’s velocity. It satisfies the performance requirement of the real world problems.     

On Beneš' bang-bang control problem

This paper treats Bene' bang-bang control problem. We present a proof of the optimality of a specific control function which differs from those proofs known in the literature and which appears to be useful for other problems too. The idea is to discretize the problem, solve the latter by means of dynamic programming and then to go to the limit.This work was supported by the Sonderforschungsbereiche 21 and 72, University of Bonn, Bonn, West Germany     

The impulse influence function for de-centralized control

Impulse influence matrix function is introduced based on that the de-centralized control analysis is analogous to the sub-structural analysis in structural mechanics. The static sub-structural analysis is analogous to the usual de-centralized control, whereas the dynamic sub-structural analysis corresponds to the de-centralized control theory. The re-     

Limit-cycle-like control for 2-dimensional discrete-time nonlinear control systems and its application to the Hénon map

In this paper, a control method that generates a desired limit-cycle-like behavior for a 2-dimensional discrete-time nonlinear control system is discussed. First, we define some notations and state the problem formulation. Next, a necessary and sufficient condition of existence of a control input that realizes a desired limit-cycle-like behavior is shown. We then derive a control algorithm to solve the problem on generating limit-cycle-like behaviors, and a modification of the algorithm is also shown. Finally, we apply the two types of algorithms to a chaotic system, the Hénon map, in order to indicate the availability of the proposed method. In addition, by using the control method, we also consider a stabilization problem for the Hénon map.     

Designs of ϕ-optimal control for second-order processes

Consider a realization of the process on the intervalT=[0,1] for functionsf ₁(t),f ₂(t),...,f _n (t) inH(R), the reproducing kernel Hilbert space with reproducing kernelR(s,t) onT×T, whereR(s,t)=E[ξ(s)ξt)] is assumed to be continuous and known. Problems of the selection of functions {f _k (t)} _k=1 ⁿ to be ϕ-optimal design are given, and an unified approach to the solutions ofD-,A-,E- andD _s-optimal design problems are discussed.     

Numerical bifurcation control of Mackey–Glass system

In this paper, a mathematical physiological model, Mackey–Glass system of a delay differential equation, is considered. With a greater delay, a periodic solution arises, which characterizes the disease of chronic granulocytic leukemia (CGL). To treat such disease, a blood transfusion feedback control is considered, from the point of view of mathematical control. By using a nonstandard finite-difference (NSFD) scheme to the control system, we obtain a numerical discrete system and analyze its Neimark–Sacker and fold bifurcations. The results imply that the condition of the illness could be relieved by transfusing blood to the patient, if the control is a delay control. Finally, the effectiveness of the control are illustrated by several numerical simulations.     

Yield management for privatised air traffic control?

Air Traffic Control (ATC) ensures the safe and expeditious passage of all aircraft in controlled airspace, usually, airspace above 5000–7000?ft or around main airports. Serious airspace congestion has led to demands for more cost-effective ATC and to plans to privatise air traffic services in the UK. These trends offer numerous opportunities to apply Operational Research in the 21st century, especially to improve information management. In a context of ATC privatisation, this paper investigates how applicable yield management (YM) policies are to European ATC. Necessary conditions for the use of YM are examined and difficulties in using YM for ATC are identified. Following this, possible YM solutions are outlined. Finally, the implications of YM for airspace users are addressed.     

Markov–Dubins path via optimal control theory

Markov–Dubins path is the shortest planar curve joining two points with prescribed tangents, with a specified bound on its curvature. Its structure, as proved by Dubins in 1957, nearly 70 years after Markov posed the problem of finding it, is elegantly simple: a selection of at most three arcs are concatenated, each of which is either a circular arc of maximum (prescribed) curvature or a straight line. The Markov–Dubins problem and its variants have since been extensively studied in practical and theoretical settings. A reformulation of the Markov–Dubins problem as an optimal control problem was subsequently studied by various researchers using the Pontryagin maximum principle and additional techniques, to reproduce Dubins’ result. In the present paper, we study the same reformulation, and apply the maximum principle, with new insights, to derive Dubins’ result again. We prove that abnormal control solutions do exist. We characterize these solutions, which were not studied adequately in the literature previously, as a concatenation of at most two circular arcs and show that they are also solutions of the normal problem. Moreover, we prove that any feasible path of the types mentioned in Dubins’ result is a stationary solution, i.e., that it satisfies the Pontryagin maximum principle. We propose a numerical method for computing Markov–Dubins path. We illustrate the theory and the numerical approach by three qualitatively different examples.     

Adaptive optimal control of Signorini’s problem

In this article, we present a-posteriori error estimations in context of optimal control of contact problems; in particular of Signorini’s problem. Due to the contact side-condition, the solution operator of the underlying variational inequality is not differentiable, yet we want to apply Newton’s method. Therefore, the non-smooth problem is regularized by penalization and afterwards discretized by finite elements. We derive optimality systems for the regularized formulation in the continuous as well as in the discrete case. This is done explicitly for Signorini’s contact problem, which covers linear elasticity and linearized surface contact conditions. The latter creates the need for treating trace-operations carefully, especially in contrast to obstacle contact conditions, which exert in the domain. Based on the dual weighted residual method and these optimality systems, we deduce error representations for the regularization, discretization and numerical errors. Those representations are further developed into error estimators. The resulting error estimator for regularization error is defined only in the contact area. Therefore its computational cost is especially low for Signorini’s contact problem. Finally, we utilize the estimators in an adaptive refinement strategy balancing regularization and discretization errors. Numerical results substantiate the theoretical findings. We present different examples concerning Signorini’s problem in two and three dimensions.     

Eigenvalue control for a Finsler–Laplace operator

Using the definition of a Finsler–Laplacian given by the first author, we show that two bi-Lipschitz Finsler metrics have a controlled spectrum. We deduce from that several generalizations of Riemannian results. In particular, we show that the spectrum on Finsler surfaces is controlled above by a constant depending on the topology of the surface and on the quasireversibility constant of the metric. In contrast to Riemannian geometry, we then give examples of highly non-reversible metrics on surfaces with arbitrarily large first eigenvalue.