Game-theoretic coupled riccati equations associated to controlled linear differential systems with jump markov perturbations

In this paper we investigate several properties of the stabilizing solution of a class of systems of Riccati type differential equations with indefinite sign associated to controlled systems described by differential equations with Markovian jumping.

We show that the existence of a bounded on R ₊ and stabilizing solution for this class of systems of Riccati type differential equations is equivalent to the solvability of a control-theoretic problem, namely disturbance attenuation problem.

If the coefficients of the considered system are theta;-periodic functions then the stabilizing solution is also theta;-periodic and if the coefficients are asymptotic almost periodic functions, then the stabilizing solution is also asymptotic almost periodic and its almost periodic component is a stabilizing solution for a system of Riccati type differential equations defined on the whole real axis. One proves also that the existence of a stabilizing and bounded on R ₊ solution of a system of Riccati differential equations with indefinite sign is equivalent to the existence of a solution to a corresponding system of matrix inequalities. Finally, a minimality property of the stabilizing solution is derived.     

INFINITE ELEMENT METHOD FOR PROBLEMS ON UNBOUNDED AND MULTIPLY CONNECTED DOMAINS

1 IntroductionDifferent kinds of numerical n1etl1ods llavc been apPlied to so1lle proble1l1s o11 exteriordolllai11s successf1lly, e.g. tlle bou11dary element metl1od, the absorbillg boundary condi-tion lnethod, the spectrunl 11letllod, a11d the i11fillite eleIue1lt 1nethod. Tlle il1finite elementllletllod llas beell applied to tl1e Laplace equatioll['], the Stokes equation[']['], the plane elastic-ity systeln['1, alld the Heln1ho1tz equatioll[n. We study tl1e infinite ele111ent llletl1od tbr…     

Initial value methods in the theory of Fredholm integral equations III. The nonlinear eigenvalue problem

In this paper, it is shown that the method introduced recently by Kalaba and Scott for solving nonlinear characteristic value problems for integral operators is equivalent to a nonlinear characteristic value problem for a matrix. This is demonstrated by finding the explicit solution to the set of differential equations used by them. The results generalize those proved recently by the author for the linear problem.     

Equations for the Missing Boundary Values in the Hamiltonian Formulation of Optimal Control Problems

Partial differential equations for the unknown final state and initial costate arising in the Hamiltonian formulation of regular optimal control problems with a quadratic final penalty are found. It is shown that the missing boundary conditions for Hamilton’s canonical ordinary differential equations satisfy a system of first-order quasilinear vector partial differential equations (PDEs), when the functional dependence of the H-optimal control in phase-space variables is explicitly known. Their solutions are computed in the context of nonlinear systems with ℝ ⁿ -valued states. No special restrictions are imposed on the form of the Lagrangian cost term. Having calculated the initial values of the costates, the optimal control can then be constructed from on-line integration of the corresponding 2n-dimensional Hamilton ordinary differential equations (ODEs). The off-line procedure requires finding two auxiliary n×n matrices that generalize those appearing in the solution of the differential Riccati equation (DRE) associated with the linear-quadratic regulator (LQR) problem. In all equations, the independent variables are the finite time-horizon duration T and the final-penalty matrix coefficient S, so their solutions give information on a whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage from the symplectic structure of the Hamiltonian formalism, which allows one to reformulate Bellman’s conjectures concerning the “invariant-embedding” methodology for two-point boundary-value problems. Results for LQR problems are tested against solutions of the associated differential Riccati equation, and the attributes of the two approaches are illustrated and discussed. Also, nonlinear problems are numerically solved and compared against those obtained by using shooting techniques.     

Real valued iterative methods for solving complex symmetric linear systems

Complex valued systems of equations with a matrix R + 1S where R and S are real valued arise in many applications. A preconditioned iterative solution method is presented when R and S are symmetric positive semi‐definite and at least one of R, S is positive definite. The condition number of the preconditioned matrix is bounded above by 2, so only very few iterations are required. Applications when solving matrix polynomial equation systems, linear systems of ordinary differential equations, and using time‐stepping integration schemes based on Padé approximation for parabolic and hyperbolic problems are also discussed. Numerical comparisons show that the proposed real valued method is much faster than the iterative complex symmetric QMR method. Copyright © 2000 John Wiley & Sons, Ltd.     

Problem without initial conditions for linear and almost linear degenerate operator differential equations

We study the problem without initial conditions for linear and almost linear degenerate operator differential equations in Banach spaces. The uniqueness of a solution of this problem is proved in the classes of bounded functions and functions with exponential behavior as t → –∞. We also establish sufficient conditions for initial data under which there exists a solution of the considered problem in the class of functions with exponential behavior at infinity.     

Existence and stability of a summable generalized solution to a mathematical model of a catalytic fluidized bed process

For a system of first-order partial differential equations describing a catalytic process in a fluidized bed, we consider a mixed problem in the half-strip 0 ≤ x ≤ h, t ≥ 0. We prove the existence and uniqueness of a bounded summable generalized solution and study its stability. We prove the stabilization as t → ∞ of the values of some physically meaningful functionals of the solution.     

Finite energy solutions of self‐adjoint elliptic mixed boundary value problems

This paper describes existence, uniqueness and special eigenfunction representations of H¹‐solutions of second order, self‐adjoint, elliptic equations with both interior and boundary source terms. The equations are posed on bounded regions with Dirichlet conditions on part of the boundary and Neumann conditions on the complement. The system is decomposed into separate problems defined on orthogonal subspaces of H¹(Ω). One problem involves the equation with the interior source term and the Neumann data. The other problem just involves the homogeneous equation with Dirichlet data. Spectral representations of the solution operators for each of these problems are found. The solutions are described using bases that are, respectively, eigenfunctions of the differential operator with mixed null boundary conditions, and certain mixed Steklov eigenfunctions. These series converge strongly in H¹(Ω). Necessary and sufficient conditions for the Dirichlet part of the boundary data to have a finite energy extension are described. The solutions for a problem that models a cylindrical capacitor is found explicitly. Copyright © 2009 John Wiley & Sons, Ltd.     

Edge Green’s functions on a branched surface. Asymptotics of solutions of coordinate and spectral equations

The problems of diffraction by a slit or a strip having ideal boundary conditions, and some other problems, can be reduced to the problem of wave propagation on a multisheet surface by applying the method of reflections. Further simplifications of the problem can be achieved by applying an embedding formula. As a result, the solution of the problem with a plane wave incidence becomes expressed in terms of the edge Green’s functions, i.e., in terms of the fields generated by dipole sources localized at branchpoints of the surface. The present paper is devoted to finding the edge Green’s functions. For this problem, two sets of differential equations, namely, the coordinate and spectral equations, are used. The properties of solutions of these equations are studied. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 233–256.     

On the solution of boundary value problems with nonseparated multipoint and integral conditions

We suggest a numerical method for solving systems of linear nonautonomous ordinary differential equations with nonseparated multipoint and integral conditions. By using this method, which is based on the operation of convolution of integral conditions into local ones, one can reduce the solution of the original problem to the solution of a Cauchy problem for systems of ordinary differential equations and linear algebraic equations. We establish bounded linear growth of the error of the suggested numerical schemes. Numerical experiments were carried out for specially constructed test problems.     

On sinc discretization and banded preconditioning for linear third‐order ordinary differential equations

Some draining or coating fluid‐flow problems and problems concerning the flow of thin films of viscous fluid with a free surface can be described by third‐order ordinary differential equations (ODEs). In this paper, we solve the boundary value problems of such equations by sinc discretization and prove that the discrete solutions converge to the true solutions of the ODEs exponentially. The discrete solution is determined by a linear system with the coefficient matrix being a combination of Toeplitz and diagonal matrices. The system can be effectively solved by Krylov subspace iteration methods, such as GMRES, preconditioned by banded matrices. We demonstrate that the eigenvalues of the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the linear system. Numerical examples are given to illustrate the effective performance of our method. Copyright © 2010 John Wiley & Sons, Ltd.     

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]).      

Transmutations and irregular singular points

We consider transumtations for a class of problems in partial differential equations where the underlying equation, involving two assignable parameters, is an associated ordinary differential equation with an irregular singular point. An integral formula for the solution of this associated problem, valid for negative values of a timelike variable t, permits relating the solution of the problems in partial differential equations to be bounded or slow groth solutions of generalized heat problems. Applications of the formulas are made to Cauchy and boundary type problems.     

On solvability of non-linear semi-periodic boundary-value problem for system of hyperbolic equations

We study a non-linear semi-periodic boundary-value problem for a system of hyperbolic equations with mixed derivative. At that, the semi-periodic boundary-value problem for a system of hyperbolic equations is reduced to an equivalent problem, consisting of a family of periodic boundary-value problems for ordinary differential equations and functional relation. When solving a family of periodic boundary-value problems of ordinary differential equations we use the method of parameterization. This approach allowed to establish sufficient conditions for the existence of an isolated solution of non-linear semi-periodic boundary-value problem for a system of hyperbolic equations.     

Quaternionic Monge-Ampère equations

The main result of this article is the existence and uniqueness of the solution of the Dirichlet problem for quaternionic Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in ℍ ⁿ . We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].     

On the well-posedness of periodic problems for the system of hyperbolic equations with finite time delay

A periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well-posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.     

Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

Quantum stochastic differential equations of the form

Synthesis of controllers for damping oscillations in dynamical systems under uncertainty

We review the modern approaches to the synthesis of robust H _∞ controllers that ensure optimal damping of oscillations in dynamical systems under uncertainty. In the synthesis method based on Riccati equations, these many-parameter equations can be solved only when the parameters are contained in a bounded parallelepiped with given boundaries. The synthesis of a robust H _∞ output control for systems with unknown bounded parameters is reducible to the solution of an optimization problem constrained by a system of linear matrix inequalities. The proposed controller synthesis algorithms are implemented using standard MATLAB procedures. The efficiency of the proposed methods and algorithms is demonstrated in application to optimal damping of oscillations in a parametrically excited pendulum. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 87–104, 2004.     

A priori estimates for solutions of the first initial boundary-value problem for systems of fully nonlinear partial differential equations

We prove a priori estimates for a solution of the first initial boundary-value problem for a system of fully nonlinear partial differential equations (PDE) in a bounded domain. In the proof, we reduce the initial boundary-value problem to a problem on a manifold without boundary and then reduce the resulting system on the manifold to a scalar equation on the total space of the corresponding bundle over the manifold. St. Petersburg Architecture Building University, St. Petersburg. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 3, pp. 338–363, March, 1997.     

Differential equations for the analytic singular value decomposition of a matrix

Summary This paper is concerned with finding a smooth singular value decomposition for a matrix which is smoothly dependent on a parameter. A previous approach to this problem was based on minimisation techniques, here, in contrast, a system of ordinary differential equations is derived for the decomposition. It is shown that the numerical solution of an initial value problem associated with these differential equations provides a feasible approach to the solution of this problem. Particular consideration is given to the situation which arises with equal modulus singular values which lead to indeterminacies in the evaluations needed for the numerical solution. Examples which illustrate the behaviour of the method are included.