Shin-ichi Nakagiri

Optimal Control of Linear Retarded Systems in Banach Spaces

This paper deals with standard optimal control problems, namely, the fixed time integral convex cost problem and the time optimal control problem for linear retarded systems in Banach spaces. For the basis of optimal control theory the fundamental solution is constructed and a variation of constant formula of (mild) solutions is established. After the controlled system description and the formulation of optimal control problems are given, the retarded adjoint system is introduced. For the integral convex cost problem two existence theorems of optimal controls and necessary conditions of optimality are given. These conditions are precisely characterized by the solution of retarded adjoint system. The “pointwise” maximum principle for time varying control domain is derived from the optimality conditions. The bang-bang principle is also established for the terminal value cost problem under some regularity condition of the adjoint system. For the time optimal control problem to a target set an existence theorem is shown. In the case where the target set has interior, the maximum principle and the bang-bang principle are established for the time optimal control. Finally, a convergence theorem of time optimal controls to a point target set is given. This paper also contains illustrative examples which give technologically important control problems.

Controllability and observability of linear retarded systems in Banach spaces

Abstract Controllability and observability are studied for linear retarded systems in a Banach space X. Both the X-exact and the X-approximate controllability are considered and a number of necessary and sufficient conditions in terms of the coefficient operators and/or the fundamental solution are presented. Various corresponding concepts for observability are introduced and the duality between controllability and observability is clarified. An applicable condition for the X-approximate controllability and the X*-observability is established under the assumptions that the control is a finite dimensional one and the system of generalized eigenspaces is projectively complete. The condition is stated in terms of eigenvectors and controllers.

Stability of a system of Volterra integro-differential equations

Using new and known forms of Lyapunov functionals, this paper proposes new stability criteria for a system of Volterra integro-differential equations.

On the fundamental solution of delay-differential equations in Banach spaces

The study of (1.2) is rather classic and a great number of research papers and monographs exist (see [5, 9, 10, 15, 161 and their references). In these works, various types of existence, uniqueness, differentiability and continuous dependence theorems are established on the basis of the construction and the representation of the semi-group T(t) relating to (1.2). Our purpose here is to give two representations of the fundamental solution of (1.1) in terms of Z(t) and A, and establish a variation of constants formula for (1.1). Such expressions are useful to obtain the fundamental theorems for (1.1) and some system theoretical results [ 11, 121. We briefly explain the content of this paper. In Section 2, the notations and notions used throughout the paper and the system description are given. A definition of the mild solution is also given in Section 2 and the treatment is based on the perturbation theory (i.e., Eq. (1.1) is considered as a pertur349 0022-039618 l/090349-20

Collision avoidance in a two-point system via Liapunov's second method

02.00/O

IDENTIFICATION OF CONSTANT PARAMETERS IN PERTURBED SINE-GORDON EQUATIONS

The geometric problem of finding a path for a moving solid among other solid obstacles is well known in robotics in the area of real-time obstacle avoidance for manipulators and mobile robots. In this paper, a solution to the problem, also known as the findpath problem, is provided via the second or direct method of Liapunov. The method is used to construct control functions for the collision avoidance between two point masses which are required to move to designated areas or targets located in the horizontal plane. Two new results are presented. The first result opens up the possibility of analysing in a more effective manner the dynamics of more than two point masses. The second new result addresses, via generalized control functions, important collision avoidance issues which are (1) improving collision avoidance between objects, (2) obtaining low control inputs for collision avoidance and convergence to targets, and (3) having the best time to reach a target safely.

WEAK SOLUTIONS OF THE EQUATION OF MOTION OF MEMBRANE WITH STRONG VISCOSITY

We study the identiflcation problems of constant pa- rameters appearing in the perturbed sine-Gordon equation with the Neumann boundary condition. The existence of optimal parame- ters is proved, and necessary conditions are established for several types of observations by utilizing quadratic optimal control theory due to Lions (13).

Identifiability of stiffness and damping coefficients in euler-bernoulli beam equations with kelvin-voigt damping

Abstract. We study the equation of a membrane with strong viscosity.Based on the variational formulation corresponding to the suitable func-tion space setting, we have proved the fundamental results on existence,uniqueness and continuous dependence on data of weak solutions. 1. IntroductionA freely °exible stretched fllm is called a membrane. Let › be an openbounded set of R n with the smooth boundary i. We set Q = (0 ;T ) £ › ; § = (0 ;T ) £ i for T > 0 : The nonlinear equation of the longitudinal motionof a vibrating membrane surrounding › is described by the following Dirichletboundary value problem:(1.1) @ 2 y@t 2 i div‡ ry p1+ jryj 2 ·= 0 in Q; with(1.2) y = 0 on § ;y (0 ;x ) = y 0 ( x ) ;@y@t (1.3) (0 ;x ) = y 1 ( x ) in › ;where y is the height of the membrane. Then the reasonable physical candidatefor the potential energy is the surface area of h = y ( x ) ; x 2 › ; since energy isstored in the membrane when it is stretched. Equation (1.1) is derived as theEuler-Lagrange equation of the action integralZ

Pointwise Completeness and Degeneracy of Functional Differential Equations in Banach Spaces. I. General Time Delays

This paper studies the identifiability problem of stiffness and damping parameters and initial value problems in Euler-Bernoulli beam equations with Kelvin-Voigt damping. Using abstract evolution equation approach, we establish conditions for the identifiability of these parameters in beam equations. Analogous identifiability conditions for beam equations without damping are also established.

Quantum numerical control for free elementary particle

Abstract A general theory for pointwise completeness and degeneracy of functional differential equations in infinite dimensional spaces is presented. For the basis of the theory, a fundamental solution and a retarded resolvent are introduced and the fact that the Laplace transform of the fundamental solution is the retarded resolvent is proved. The concepts of exact and approximate pointwise completeness are defined by the (attainable) sets of all possible mild solutions of the equations and are investigated in the framework of linear operator theory. A necessary and sufficient condition and a negative result for exact pointwise completeness are given. The degenerate space is defined by the orthogonal complement of the attainable set and various characterizations of the space in terms of fundamental solution and/or coefficient operators are established with some examples. Interesting characterizations of the full degenerate set in terms of retarded resolvent and generalized eigenfunctions, which extend the well-known finite-dimensional degeneracy conditions, are also established.