## SEARCH

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## On the Dido Problem and Plane Isoperimetric Problems

### Acta Applicandae Mathematica (1999-07-01) 57: 287-338 , July 01, 1999

This paper is a continuation of a series of papers, dealing with contact sub-Riemannian metrics on R^{3}. We study the special case of contact metrics that correspond to isoperimetric problems on the plane. The purpose is to understand the nature of the corresponding optimal synthesis, at least locally. It is equivalent to studying the associated sub-Riemannian spheres of small radius. It appears that the case of generic isoperimetric problems falls down in the category of generic sub-Riemannian metrics that we studied in our previous papers (although, there is a certain symmetry). Thanks to the classification of spheres, conjugate-loci and cut-loci, done in those papers, we conclude immediately. On the contrary, for the Dido problem on a 2-d Riemannian manifold (i.e. the problem of minimizing length, for a prescribed area), these results do not apply. Therefore, we study in details this special case, for which we solve the problem generically (again, for generic cases, we compute the conjugate loci, cut loci, and the shape of small sub-Riemannian spheres, with their singularities). In an addendum, we say a few words about: (1) the singularities that can appear in general for the Dido problem, and (2) the motion of particles in a nonvanishing constant magnetic field.

## Optimal control problem governed by semilinear parabolic equation and its algorithm

### Acta Mathematicae Applicatae Sinica, English Series (2008-01-01) 24: 29-40 , January 01, 2008

In this paper, an optimal control problem governed by semilinear parabolic equation which involves the control variable acting on forcing term and coefficients appearing in the higher order derivative terms is formulated and analyzed. The strong variation method, due originally to Mayne et al to solve the optimal control problem of a lumped parameter system, is extended to solve an optimal control problem governed by semilinear parabolic equation, a necessary condition is obtained, the strong variation algorithm for this optimal control problem is presented, and the corresponding convergence result of the algorithm is verified.

## Allocation of an indivisible resource: Optimal control and prices

### Journal of Applied and Industrial Mathematics (2010-07-01) 4: 380-388 , July 01, 2010

Under study is a dynamical system of economics with discrete time whose states at each moment correspond to nonnegative integer points of a two-dimensional vector space. There are two types of products and two facilities each of which can change the state of the system by random integer vectors with different collections of probabilities. By a control we understand the choice at each moment of time of one available collection of probabilities. The goal of control is to minimize the probability of leaving the positive quadrant. The question is considered of the existence of some prices that agree with an optimal control.

## Optimization of a Class of Nonlinear Dynamic Systems: New Efficient Method without Lagrange Multipliers

### Journal of Optimization Theory and Applications (1998-04-01) 97: 11-28 , April 01, 1998

This paper deals with optimization of a class of nonlinear dynamic systems with *n* states and *m* control inputs commanded to move between two fixed states in a prescribed time. Using conventional procedures with Lagrange multipliers, it is well known that the optimal trajectory is the solution of a two-point boundary-value problem. In this paper, a new procedure for dynamic optimization is presented which relies on tools of feedback linearization to transform nonlinear dynamic systems into linear systems. In this new form, the states and controls can be written as higher derivatives of a subset of the states. Using this new form, it is possible to change constrained dynamic optimization problems into unconstrained problems. The necessary conditions for optimality are then solved efficiently using weighted residual methods.

## Based on wavelet analysis to optimal control of motion planning of space manipulator

### Applied Mathematics and Mechanics (2000-10-01) 21: 1161-1168 , October 01, 2000

The optimal control problem of nonholonomic motion planning of space manipulator was discussed. Utilizing the method of wavelet analysis, the discrete orthogonal wavelets were introduced to solve the optimal control problem, the classical Fourier basic functions were replaced by the wavelet expansion approximation. A numerical algorithm of optimal control was proposed based on wavelet analysis. The numerical simulation shows, the method is effective for nonholonomic motion planning of space manitulator.

## Numerical solution of the nonstationary Stokes system by methods of adjoint-equation theory and optimal control theory

### Computational Mathematics and Mathematical Physics (2007-07-01) 47: 1142-1157 , July 01, 2007

Methods in optimal control and the adjoint-equation theory are applied to the design of iterative algorithms for the numerical solution of the nonstationary Stokes system perturbed by a skew-symmetric operator. A general scheme is presented for constructing algorithms of this kind as applied to a broad class of problems. The scheme is applied to the nonstationary Stokes equations, and the convergence rate of the corresponding iterative algorithm is examined. Some numerical results are given.

## Effective Information for Offline Stochastic Feedback and Optimal Control of Dynamic Systems

### Journal of Optimization Theory and Applications (2003-02-01) 116: 283-310 , February 01, 2003

The impact of uncertain future events on decision making in a stochastic environment is modeled in this paper. Such modeling is presented for both feedback and optimal control problems. This research overcomes the difficulties of forecasting that arise when considering future information. In this paper, we seek to find the minimum amount of information (effective information) necessary to evaluating system performance offline or to optimally control a system. The existence of effective information is proved and a methodology for determining it is developed. It is also shown that ignoring information beyond the planning horizon leads to significant performance loss and may even lead to violating the constraints of a control problem.

## Investigation of the optimal control of metal solidification for a complex-geometry object in a new formulation

### Computational Mathematics and Mathematical Physics (2014-12-01) 54: 1804-1816 , December 01, 2014

New formulations of the optimal control problem for metal solidification in a furnace are proposed in the case of an object of complex geometry. The underlying mathematical model is based on a three-dimensional two-phase initial-boundary value problem of the Stefan type. The formulated problems are solved numerically with the help of gradient optimization methods. The gradient of the cost function is exactly computed by applying the fast automatic differentiation technique. The research results are described and analyzed. Some of the results are illustrated.

## Game Models for the Control of the Main Body Functional Systems and their Analysis. I

### Cybernetics and Systems Analysis (2014-01-01) 50: 68-80 , January 01, 2014

The authors propose and validate the mathematical models for the optimal control of dynamic processes in the respiratory, circulatory, erythropoiesis, and heat exchange functional systems based on the solution of multiobjective optimization problems. The intra- and intersystem interactions are analyzed for a body in highland and strenuous physical activity.

## The problem of controlling resources on network graphs as an optimal control problem

### Moscow University Computational Mathematics and Cybernetics (2014-04-01) 38: 59-63 , April 01, 2014

The problem of optimal distribution of resources dedicated to a certain complex of inter-related tasks according to the criterion of minimum execution time of all tasks is described. A reenterable (reusable) resource is considered instead of a traditional separable-type resource supposing a fixed distribution among tasks. The problem is formalized in direct static and dynamic settings. The latter is a classical performance optimal control problem. The correctness of the formalization is substantiated.