Periodic optimal control in Hilbert space

Optimal control problem governed byy=Ay + Bu, y(0)=y (T), =±1 are studied, whereA is the infinitesimal generator of a nonasymptotically stableC _o semigroup andB is a linear operator from a controller spaceU into a state spaceH. Both distributed (B L(U, H)) and boundary cases (B L(U, (D(A ^*)))) are investigated. Some applications to periodic control of wave equations are given.This work was supported by the National Science Foundation under Grant No. DMS-91-11794.     

A variational problem in Hilbert space

Let? be a Hilbert space over the field of complex numbers, with inner product (g,h). Letf be a fixed element in?, and letH be a compact, self-adjoint linear operator on?. We find the maximum value ofQ _f (u)=|(f,u)|² in the classU of elementsu in? for which (u,u)=1, (u, Hu)=0.     

Sensitivity analysis of optimization problems in Hilbert space with application to optimal control

A family of optimization problems in a Hilbert space depending on a vector parameter is considered. It is assumed that the problems have locally isolated local solutions. Both these solutions and the associated Lagrange multipliers are assumed to be locally Lipschitz continuous functions of the parameter. Moreover, the assumption of the type of strong second-order sufficient condition is satisfied.It is shown that the solutions are directionally differentiable functions of the parameter and the directional derivative is characterized. A second-order expansion of the optimal-value function is obtained. The abstract results are applied to state and control constrained optimal control problems for systems described by nonlinear ordinary differential equations with the control appearing linearly.     

Remarks on bang-bang control in a Hilbert space

The control of a linear system, whose performance index is the sum of a linear term and a quadratic term, is considered. A necessary and sufficient condition is given for the optimal control to be bang-bang, and this is used to extend and clarify the results of Refs. 1–2. As an illustration, an application to an elliptic boundary-value problem is given.This research was supported by the SFB 72 of the DFG, West Germany.     

Inverse problem for potentials of measures in a Hilbert space

For a Banach space E and a number p R, under the assumption of the convergence of the integral, one considers the potential g(a)= _E x–a ^p d(x), where is a finite Borel measure on E. One solves the problem of the determination of the measure from known values of the function g(a),a E. In the note one gives an explicit solution of the problem for an infinite-dimensional Hilbert space, which exists for p0, 2, 4,.... For certain finite-dimensional Banach spaces one solves the inverse problem with the aid of the known Levy representations for the norms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 73–77, 1989.     

Inverse problem for a sturm-liouville elliptic equation in a Hilbert space

For a second-order operator-differential equation of elliptic type, the problem is stated and solved of finding for the right-hand side of the equation n vector parameters for which a solution of a Dirichlet problem takes preassigned values at n interior points.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 11, pp. 1537–1540, November, 1991.     

Optimal control of a two-parameter system in a Hilbert space

An optimal control problem is investigated for a rather wide class of two-parameter discrete systems in a separable Hilbert space. Bibliography: 12 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 80, 1996, pp. 12–24.     

An abstract Dirichlet problem in the Hilbert space

In the present paper, we consider an abstract partial differential equation of the form $\frac{{\partial ^2 u}}{{\partial t^2 }} - \frac{{\partial ^2 u}}{{\partial x^2 }} + A\left( {x,t} \right)u = f\left( {x,t} \right)$ , where $\left\{ {A\left( {x,t} \right):\left( {x,t} \right) \in \bar G} \right\}$ is a family of linear closed operators and $\bar G = G \cup \partial G,G$ is a suitable bounded region in the (x, t)-plane with boundary?G. It is assumed thatu is given on the boundary?G. The objective of this paper is to study the considered Dirichlet problem for a wide class of operatorsA(x, t). A Dirichlet problem for non-elliptic partial differential equations of higher orders is also considered.     

Remarks on bang-bang control in Hilbert space

In this note, a natural definition of bang-bang control in Hilbert space is given, and some of the theory of the authors' paper (Ref. 1) is rebuilt upon it. An elliptic boundary-value problem illustrating the theory is given. In the last part of this note, the results of Ref. 1 are extended to nonlinear perturbations of linear operators and to homogeneous nonlinear operators.Communicated by L. CesariThis research was supported by NRC Grant No. A-4047 and NSF Grant No. GP-7445.     

On a student-related optimal control problem

In a recent paper (Ref. 1), Cheng and Teo discussed some further extensions of a student-related optimal control problem which was originally proposed by Raggettet al. (Ref. 2) and later on modified by Parlar (Ref. 3). In this paper, we treat further extensions of the problem.This paper is a modified and improved version of Ref. 4. It is based, in part, on research sponsored by NSF.     

On the problem of Hermite interpolation of operators in a Hilbert space

The problem of Hermite operator interpolation with interpolational conditions containing Gateaux higher-order differentials in arbitrary directions is investigated. A necessary and sufficient condition for solvability of this problem in a Hilbert space is established, and the set of all Hermite operator polynomials and its subset of interpolants preserving operator polynomials of the same degree are described. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 38–48.     

Metric geodesics of isometries in a Hilbert space and the extension problem

We consider the problem of finding short smooth curves of isometries in a Hilbert space . The length of a smooth curve , , is measured by means of , where denotes the usual norm of operators. The initial value problem is solved: for any isometry and each tangent vector at (which is an operator of the form with ) with norm less than or equal to , there exist curves of the form , with initial velocity , which are short along their path. These curves, which we call metric geodesics, need not be unique, and correspond to the so called extension problem considered by M.G. Krein and others: in our context, given a symmetric operator

find all possible extending to all , with . We also consider the problem of finding metric geodesics joining two given isometries and . It is well known that if there exists a continuous path joining and , then both ranges have the same codimension. We show that if this number is finite, then there exist metric geodesics joining and .

     

Relatively optimal filtering on a Hilbert space for measure driven stochastic systems

In this paper we consider linear filtering for discontinuous processes determined by stochastic differential equations on a Hilbert space driven by signed measures in addition to Brownian motion. The dynamics of the observed data is governed by a differential equation driven by a square integrable martingale (not necessarily continuous) while perturbed by a signed measure. We formulate the filtering problem as an optimization problem on the space of bounded linear operator valued functions and present necessary and sufficient conditions for optimality. Further, we prove, under the assumption of finite dimensionality of the output space, that a Kalman-like filter exists and it is explicitly determined by a Riccati type evolution equation.     

On the infimum problem of Hilbert space effects

The quantum effects for a physical system can be described by the set ε(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. The infimum problem of Hilbert space effects is to find under what condition the infimum A∧B exists for two quantum effects A and B∈ε(H). The problem has been studied in different contexts by R. Kadison, S. Gudder, M. Moreland, and T. Ando. In this note, using the method of the spectral theory of operators, we give a complete answer of the infimum problem. The characterizations of the existence of infimum A∧B for two effects A. B∈ε(H) are established.     

Sampling in a Hilbert space

An analog of the Whittaker-Shannon-Kotel'nikov sampling theorem is derived for functions with values in a separable Hilbert space. The proof uses the concept of frames and frame operators in a Hilbert space. One of the consequences of this theorem is that it allows us to derive sampling theorems associated with boundary-value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem by Kramer.