On synthesizing impulse controls and the theory of fast controls

We describe the theory of feedback control in the class of impulse-type inputs which allow higher derivatives of delta functions. We provide solutions based on Hamiltonian techniques in the dynamic programming form. Further we describe physically realizable approximations of the “ideal” impulse-type solutions by bounded functions which may also serve as “fast” feedback controls that solve the target control problem in arbitrarily small time.     

Zero-sum differential games involving impulse controls

In this paper we are concerned with zero-sum differential games with impulse controls, as well as continuous and switching controls. The motivation is optimal impulse control problems with disturbances. The main result is the existence of the value function of the game. Our approach is the theory of viscosity solutions for Hamilton-Jacobi equations.Part of this paper was done while the author was a visiting scholar at INRIA, Sophia-Antipolis, France. This work was also partially supported by the Chinese NSF, the Chinese State Education Commission Science Foundation, and the Fok Ying Tung Education Foundation.     

Synthesis of impulse and fast controls under uncertainty

Doklady Mathematics -     

Differential games with continuous,switching and impulse controls

A two-person zero-sum differential game of infinite duration with discounted payoff involving hybrid controls is studied. The minimizing player is allowed to take continuous, switching, and impulse controls whereas the maximizing player is allowed to take continuous and switching controls. By taking strategies in the sense of Elliott–Kalton, we prove the existence of value and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities.     

Approximation of strongly relaxed minimizers with ordinary delayed controls

In a recent paper, a procedure for relaxing optimal control problems with nonseparable commensurate delays in the controls was proposed. This model (strong relaxation) was shown to be proper in the sense that the minimum cost for the relaxed problem coincides with the infimum cost for the original one, and it solves some of the difficulties encountered when another procedure, based on a well-known technique for reducing a control system of this nature to a delay-free problem, is used. However, the proof of its being proper is unsatisfactory in two respects, not only from a theoretical point of view but also in order to construct numerical approximation algorithms. In this proof the use of the reduced-problem technique is still needed, so that the disadvantages of this procedure remain present. Moreover, the existence of a sequence of ordinary delayed controls which converges to a given strongly relaxed control is assured, but one may fail to exhibit it. In this paper we solve these questions by giving an explicit methodology, which is independent of the reduced-problem technique, for constructing such a sequence. Also, this constructive approach provides a new and simple proof of the fact that the space of strongly relaxed controls coincides with the weak-star closure of the space of ordinary delayed controls.     

Optimal bang-bang controls arising in a Sobolev impulse control problem

In this note the switches of optimal bang-bang controls associated with Sobolev impulse control problems are studied. The determination of the number of switches in such controls is discussed and examples are considered. Also, sequences of approximating controls arising from the variational optimality conditions are shown to converge almost everywhere to the optimal control.     

Approximation of point controls of second-order evolution equations of Sobolev type

Limit behavior of the optimal-point control associated with a Sobolev equation as inertial or viscous effect is allowed to vanish is studied. These results are also related to the approximation of a wave equation.     

Approximation of sparse controls in semilinear equations by piecewise linear functions

Semilinear elliptic optimal control problems involving the $L^1$ norm of the control in the objective are considered. A priori finite element error estimates for piecewise linear discretizations for the control and the state are proved. These are obtained by a new technique based on an appropriate discretization of the objective function. Numerical experiments confirm the convergence rates.     

Maximum principle for optimal control problems involving impulse controls with nonsmooth data

This paper is concerned with the stochastic maximum principle for impulse optimal control problems of forward–backward systems, where the coefficients of the forward part are Lipschitz continuous. The domain of the regular controls is not necessarily convex. We establish a Pontryagins maximum principle for this control problem by applying Ekelands variational principle to a sequence of approximated control problems with smooth coefficients of the initial problems.     

Systems governed by ordinary differential equations with continuous,switching and impulse controls

Optimal control problem for systems governed by ordinary differential equations with continuous, switching and impulse controls are studied. It is proved that the value function of the problem is the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman system.     

Uniform Approximation of Functions with Discrete Approximation Functionals

Hoffman, Kouri, and collaborators have calculated nonrelativistic quantum scattering amplitudes by numerically evaluating Feynman path integrals. They observed that the errors introduced by their numerical scheme were uniform in coordinate space, implying that their scheme accurately reproduces both the shape and the phase of functions. Furthermore, they observed that the size and the uniform nature of the errors were preserved when the functions were allowed to evolve in time under the action of the kinetic energy operator. In this paper it is established that these observed properties of the errors are not numerical artifacts but follow from analytical properties of a general class of approximations that include those of Hoffman, Kouri, and collaborators as a special case.     

Optimal impulse correction

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 8, pp. 1058–1064, August, 1989.     

多元周期函数的一类逼近及逼近阶估计

本文讨论了多元周期函数的一类逼近,得到一个稠密性的充要条件并构造出适当的逼近函数,给出了逼近阶的精确估计．     

论可微函数的共单调逼近和共凸逼近

对有限区间上可微函数借助于代数多项式的共单调逼近和共凸逼近的逼近度估计建立了更为精确的Jackson型不等式,扩充和改进了近期的一些结果。     

C-伪预解式的逼近

引入了C-伪预解式的概念,并讨论了它的一些性质及收敛性,最后给出了C-伪预解式收敛的一个充分条件.     

Discrete Approximation of Unbounded Operators and Approximation of their Spectra

Let E be a Banach space over and let the densely defined closed linear operator A: (A)E→E be discretely approximated by the sequence ((A_n, (A_n)))_n of operators A_n where each A_n is densely defined in the Banach space F_n. Let σ_a(A) be the approximate point spectrum of A and let σ(A_n) denote the -pseudospectrum of A_n. Generalizing our own result, we show that σ_a(A)lim inf σ(A_n)=_n ∩_kn σ(A_k) holds for every >0. We deduce that then for every compact set K lim_n dist(σ_a(A)∩K, σ_a(A_n))=0 provided there exists M>0 such that (λ−A_n)⁻¹M dist(λ, σ(A_n))⁻¹ holds for every n and every λ in the resolvent set ρ(A_n) of A_n. We finally treat the problem under which conditions σ_a(A) can be approximated from below. More precisely we investigate the problem: Under which assumptions does ∩_>0 ∩_{n kn} σ_, a(A_k)σ_a(A) hold where σ_, a(A) denotes the -approximate pseudospectrum?     

On Approximation Algorithm for Orthogonal Low-Rank Tensor Approximation

Journal of Optimization Theory and Applications - This work studies solution methods for approximating a given tensor by a sum of R rank-1 tensors with one or more of the latent factors being...     

Approximation of Pseudoresolvents

After a short introduction where a few main notions and results are recalled, we state some results connected with generated and L_∞-type pseudoresolvents. Among others, we give theorems of characterization both for L_∞-type pseudoresolvents and for their generators. The relation between L_∞-type pseudoresolvents and C₀-equicontinuous semigroups is also pointed out. The last part of the present paper is devoted to approximation of pseudoresolvents and their generators. If are generated pseudoresolvents and A_n, A their generators, we investigate conditions under which A is approximated by A_n and R is approximated by R_n, n ≥ 1. In addition, we give conditions under which a sequence of generated pseudoresolvents approximates a pseudoresolvent, and in this case we study the connection between generators. Received: February 14, 2008., Revised: June 2, 2008., Accepted: August 24, 2008.     

Averaging of impulse evolution systems

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 76–83, January, 1992.