Subspace exit properties of extremal controls

In a linear control system setting, we derive conditions under which extremal controls give rise to trajectories which exit a given subspace. An application is in determining effectiveness of linear state restraints in time-optimal problems.This work was supported by the National Research Council of Canada under Grant No. A4641.     

Min-max control of systems approximated by simple models:L 1-Type cost functionals

A control problem for actual processes, modeled by simple dynamic linear systems, is studied. A min-max approach is taken, since the problem is reduced to the control of a model, whose output is affected by an input-dependent signal constrained in norm.Particular attention is given to the features of the penalty functional, since it is desired to synthesize the min-max control by means of a feedback on the states of the model; for example, a quadratic functional does not have this property. AnL ₁-type functional is then proposed, which is not differentiable; by means of duality techniques, however, the minimization can be carried out. The min-max is obtained both by iterative algorithms, both as a function of the actual value of the model state: this function is the solution of a set of partial differential equations.This work was supported by the CENS and the Gruppo Nazionale per l'Analisi Funzionale e le sue Applicazioni, Consiglio Nazionale delle Ricerche (CNR), Roma, Italy.     

An exponential differential game which admits a simple Nash solution

This paper deals with a class ofN-person nonzero-sum differential games where the control variables enter into the state equations as well as the payoff functionals in an exponential way. Due to the structure of the game, Nash-optimal controls are easily determined. The equilibrium in open-loop controls is also a closed-loop equilibrium. An example of optimal exploitation of an exhaustible resource is presented.The helpful comments of Professor Y. C. Ho and Dipl. Ing. E. Dockner are gratefully acknowledged.     

Insensitizing controls for a class of nonlinear Ginzburg-Landau equations

This paper shows the existence of insensitizing controls for a class of nonlinear complex Ginzburg-Landau equations with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. When the nonlinearity in the equation satisfies a suitable superlinear growth condition at infinity, the existence of insensitizing controls for the corresponding semilinear Ginzburg-Landau equation is proved. Meanwhile, if the nonlinearity in the equation is only a smooth function without any additional growth condition, a local result on insensitizing controls is obtained. As usual, the problem of insensitizing controls is transformed into a suitable controllability problem for a coupled system governed by a semilinear complex Ginzburg-Landau equation and a linear one through one control. The key is to establish an observability inequality for a coupled linear Ginzburg-Landau system with one observer.     

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.     

Hamiltonian Pontryagin's Principles for Control Problems Governed by Semilinear Parabolic Equations

In this paper we study optimal control problems governed by semilinear parabolic equations. We obtain necessary optimality conditions in the form of an exact Pontryagin's minimum principle for distributed and boundary controls (which can be unbounded) and bounded initial controls. These optimality conditions are obtained thanks to new regularity results for linear and nonlinear parabolic equations. Accepted 17 March 1997     

Optimal Control of Uniformly Heated Granular Fluids in Linear Response

We present a detailed analytical investigation of the optimal control of uniformly heated granular gases in the linear regime. The intensity of the stochastic driving is therefore assumed to be bounded between two values that are close, which limits the possible values of the granular temperature to a correspondingly small interval. Specifically, we are interested in minimising the connection time between the non-equilibrium steady states (NESSs) for two different values of the granular temperature by controlling the time dependence of the driving intensity. The closeness of the initial and target NESSs make it possible to linearise the evolution equations and rigorously—from a mathematical point of view—prove that the optimal controls are of bang-bang type, with only one switching in the first Sonine approximation. We also look into the dependence of the optimal connection time on the bounds of the driving intensity. Moreover, the limits of validity of the linear regime are investigated.     

Volterra积分系统的Minimax最优控制

本文对Volterra积分系统的minimax最优控制问题,建立了相应的Pontryagin最大值原理.