Stability estimates in identification problems for the convection-diffusion-reaction equation

Identification problems for the stationary convection-diffusion-reaction equation in a bounded domain with a Dirichlet condition imposed on the boundary of the domain are studied. By applying an optimization method, these problems are reduced to inverse extremum problems in which the variable diffusivity and the volume density of substance sources are used as control functions. Their solvability is proved for an arbitrary weakly lower semicontinuous cost functional and particular cost functionals. An analysis of the optimality system is used to establish sufficient conditions on the input data under which the solutions of particular extremum problems are unique and stable with respect to small perturbations in the cost functional and in one of the functions involved in the boundary value problem.     

Conditions for the discovery of solution horizons

We present necessary and sufficient conditions for discrete infinite horizon optimization problems with unique solutions to be solvable. These problems can be equivalently viewed as the task of finding a shortest path in an infinite directed network. We provide general forward algorithms with stopping rules for their solution. The key condition required is that of weak reachability, which roughly requires that for any sequence of nodes or states, it must be possible from optimal states to reach states close in cost to states along this sequence. Moreover the costs to reach these states must converge to zero. Applications are considered in optimal search, undiscounted Markov decision processes, and deterministic infinite horizon optimization.This work was supported in part by NSF Grant ECS-8700836 to The University of Michigan.     

Constrained extremum problems with infinite dimensional image. Selection and saddle point

The paper deals with Image Space Analysis for constrained extremum problems having infinite dimensional image. It is shown that the introduction of selection for point- to-set maps and of quasi-multipliers allows one to establish sufficient optimality conditions for problems, where the classic ones fail.      

Optimality conditions for state constrained nonlinear control problems

The problem of optimal control of nonlinear control and state constrained control problems, where the state constraint may involve differential operators and the cost functionals may be nonsmooth, is studied. For this class of problems, necessary optimality conditions using techniques from infinite dimensional optimization theory adapted to the framework of control problems are derived. It is shown that the underlying structure admits a considerable relaxation of the classical constraint qualifications. The theory then is applied to examples of various nonlinear elliptic equations and state constraints.     

Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability

In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential Riccati equation (BDRE) associated with these problems (see [2], for finite dimensional stochastic equations or [21], for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see [10], [18]). Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive, we establish that BDRE has a unique, uniformly positive, bounded on ℝ ₊ and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known [18] that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see [10]).      

Extremal points and optimal solutions for general capacity problems

This paper studies the infinite dimensional linear programming problems in the integration type. The variable is taken in the space of bounded regular Borel measures on compact Hausdorff spaces. It will find an optimal measure for a constrained optimization problem, namely a capacity problem. Relations between extremal points of the feasible region and optimal solutions of the optimization problem are investigated. The necessary/sufficient conditions for a measure to be optimal are established. The algorithm for optimal solution of the general capacity problem onX = Y = [0, 1] is formulated.     

Krein空间中无穷维Hamilton算子的极大确定不变子空间的存在性问题

本文研究了不定度规空间空间中的无穷维Hamilton算子.利用Plus算子存在极大不变子空间的性质,获得了无穷维Hamilton算子在Krein空间中存在极大确定不变子空间的充分条件.     

Vortex control in channel flows using translational invariant cost functionals

The use of translation invariant cost functionals for the reduction of vortices in the context of shape optimization of fluid flow domain is investigated. Analytical expressions for the shape design sensitivity involving different cost functionals are derived. Channel flow problems with a bump and an obstacle as possible control boundaries are taken as test examples. Numerical results are provided in various graphical forms for relatively low Reynolds numbers. Striking differences are found for the optimal shapes corresponding to the different cost functionals, which constitute different quantification of a vortex.     

A comparison of alternative c-conjugate dual problems in infinite convex optimization

In this work, we obtain a Fenchel–Lagrange dual problem for an infinite dimensional optimization primal one, via perturbational approach and using a conjugation scheme called c-conjugation instead of classical Fenchel conjugation. This scheme is based on the generalized convex conjugation theory. We analyse some inequalities between the optimal values of Fenchel, Lagrange and Fenchel–Lagrange dual problems and we establish sufficient conditions under which they are equal. Examples where such inequalities are strictly fulfilled are provided. Finally, we study the relations between the optimal solutions and the solvability of the three mentioned dual problems.     

Vortex control of instationary channel flows using translation invariant cost functionals

The use of translation invariant cost functionals for the reduction of vortices in the context of shape optimization of fluid flow domains is investigated. Analytical expressions for the shape design sensitivity involving different cost functionals are derived. Instationary channel flow problems with a bump and an obstacle as possible control boundaries are taken as test examples. Numerical results are provided in various graphical forms for relatively low Reynolds numbers. Striking differences are found for the optimal shapes corresponding to the different cost functionals, which constitute different quantification of a vortex.     

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.     

On the Finite-dimensional Optimal Approximations to the Infinite dimensional Linear Operators

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＨａｎｄＫｂｅｒｅａｌｏｒｃｏｍｐｌｅｘｉｎｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌＨｉｌｂｅｒｓｐａｃｅ．Ａ（ｂｏｕｎｄｅｄｌｉｎｅａｒ）ｏｐ－ｅｒａｔｏｒＡ：Ｈ→Ｋｉｓｓａｉｄｔｏｂｅｉｎｆｉｎｉｔｅ－ｄｉｍｅｎｓｉｏｎａｌ（ｏｒ,ｆｉｎｉｔｅ－ｄｉｍｅｎｓｉｏｎａｌ）ｉｆｔｈｅｒａｎｇｅｏｆＡｉｓｉｎｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌ（ｏｒ,ｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌ）．Ａｉｓｓａｉｄｔｏｂｅｎ－ｄｉｍｅｎｓｉｏｎａｌｉｆｔｈｅｒａｎｇｅｏｆＡｉｓｎ…     

The problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures

We obtain a linear programming characterization for the minimum cost associated with finite dimensional reflected optimal control problems. In order to describe the value functions, we employ an infinite dimensional dual formulation instead of using the characterization via Hamilton-Jacobi partial differential equations. In this paper we consider control problems with both infinite and finite horizons. The reflection is given by the normal cone to a proximal retract set.     

Quadratic Control for Stochastic Systems Defined by Evolution Operators and Square Integrable Martingales

The infinite dimensional version of the linear quadratic cost control problem is studied by Curtain and Pritchard [2], Gibson [5] by using Riccati integral equations, instead of differential equations. In the present paper the corresponding stochastic case over a finite horizon is considered. The stochastic perturbations are given by Hilbert valued square integrable martingales and it is shown that the deterministic optimal feedback control is also optimal in the stochastic case. Sufficient conditions are given for the convergence of approximate solutions of optimal control problems.     

基于参数李雅普诺夫函数的鲁棒预测控制器

针对多包描述的不确定系统,提出一种新的鲁棒约束预测控制器.离线设计时引入参数Lyapunov函数以减少单一Lyapunov函数设计时的保守性,得到多包系统Worst-case情况下性能最优的不变集,在线求解多包系统无穷时域性能指标的min-max优化问题.设计采用了时变的终端约束集,扩大了初始可行域,而且能够获得较优的控制性能.仿真结果验证了该方法的有效性.     

Decomposing Ends of Locally Finite Graphs

An important invariant of translations of infinite locally finite graphs is that of a direction as introduced by Halin . This invariant gives not much information if the translation is not a proper one. A new refined concept of directions is investigated. A double ray D of a graph X is said to be metric, if the distance metrics in D and X on V(D) are equivalent. It is called geodesic, if these metrics are equal. The translations leaving some metric double ray invariant are characterized. Using a result of Polat and Watkins , we characterize the translations leaving some geodesic double ray invariant.     

Stability estimates for the solutions to inverse extremal problems for the Helmholtz equation

The inverse problems are under study for the Helmholtz equation describing acoustic scattering at a three-dimensional inclusion. Some optimization method reduces these problems to the inverse extremum problems with variable refraction index and boundary source density as controls. We prove that these problems are solvable and derive the optimality systems that describe necessary optimality conditions. Analysis of the optimality systems leads us to some sufficient conditions on the input data ensuring the uniqueness and stability of optimal solutions.     

Square indefinite LQ-problem: Existence of a unique solution

In this paper, we consider discrete-time systems. We study conditions under which there is a unique control that minimizes a general quadratic cost functional. The system considered is described by a linear time-invariant recurrence equation in which the number of inputs equals the number of states. The cost functional differs from the usual one considered in optimal control theory, in the sense that we do not assume that the weight matrices considered are semipositive definite. For both a finite planning horizon and an infinite horizon, necessary and sufficient solvability conditions are given. Furthermore, necessary and sufficient conditions are derived for the existence of a solution for an arbitrary finite planning horizon.The author dedicates this paper to the memory of his late grandfather Jacob Oosterwold.     

A Harnack-type inequality for non-symmetric Markov semigroups

For a strong Feller and irreducible Markov semigroup on a locally compact Polish space, the Harnack-type inequality (1.1) holds if and only if the semigroup has a unique invariant probability measure and is ultracontractive. Moreover, new sufficient conditions for this inequality to hold, as well as upper bound estimates of the underlying constant, are presented for diffusion semigroups on Riemannian manifolds.     

On the approximation of infinite optimization problems with an application to optimal control problems

This paper is concerned with approximations to infinite optimization problems in Banach spaces. Under the assumption of a first order necessary and a second order sufficient optimality condition we derive convergence results for the optimal solutions and the optimal values of the approximating problems. An application to finite difference approximations of nonlinear optimal control problems with state constraints is given.