Representation of a solution for a system of linear inhomogeneous two-dimensional difference equations of fractional order

Fractional Orders ◽

Two Parameter ◽

The Pontryagin Maximum Principle

One linear inhomogeneous two-parameter discrete fractional system is considered, and the boundary condition is a solution of an analogue of the Cauchy problem for a linear ordinary difference equation. Equation coefficients are given by discrete matrix functions. By introducing an analogue of the Riemann matrix, representations of solutions of the considered boundary value problem are obtained. Note that the result obtained plays an essential role in the linear case for establishing a necessary and sufficient optimality condition in the form of the Pontryagin maximum principle, and also in the general case for studying special control in discrete optimal control problems for systems of 2D fractional orders.

Solving the Hamilton-Jacobi-Bellman equation of stochastic control by a semigroup perturbation method

Advances in Applied Probability ◽

10.1017/s0001867800022084 ◽

1984 ◽

Vol 16 (1) ◽

pp. 16-16

Author(s):

Domokos Vermes

Keyword(s):

Bellman Equation ◽

Value Function ◽

Sufficient Optimality Condition ◽

Hamilton Jacobi Bellman Equation ◽

Deterministic Processes ◽

Random Jumps ◽

Hamilton Jacobi Bellman ◽

The Value Function ◽

Deterministic Trajectories

We consider the optimal control of deterministic processes with countably many (non-accumulating) random iumps. A necessary and sufficient optimality condition can be given in the form of a Hamilton-jacobi-Bellman equation which is a functionaldifferential equation with boundary conditions in the case considered. Its solution, the value function, is continuously differentiable along the deterministic trajectories if. the random jumps only are controllable and it can be represented as a supremum of smooth subsolutions in the general case, i.e. when both the deterministic motion and the random jumps are controlled (cf. the survey by M. H. A. Davis (p.14)).

Approximate Controllability of Partial Fractional Neutral Integro-Differential Inclusions With Infinite Delay in Hilbert Spaces

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4030533 ◽

2016 ◽

Vol 11 (1) ◽

Cited By ~ 1

Author(s):

Zuomao Yan ◽

Hongwu Zhang

Keyword(s):

Hilbert Spaces ◽

Differential Inclusions ◽

Approximate Controllability ◽

Infinite Delay ◽

Sufficient Conditions ◽

Multivalued Operators ◽

Approximately Controllable ◽

Fractional System ◽

The Fixed Point Theorem ◽

We study the approximate controllability of a class of fractional partial neutral integro-differential inclusions with infinite delay in Hilbert spaces. By using the analytic α-resolvent operator and the fixed point theorem for discontinuous multivalued operators due to Dhage, a new set of necessary and sufficient conditions are formulated which guarantee the approximate controllability of the nonlinear fractional system. The results are obtained under the assumption that the associated linear system is approximately controllable. An example is provided to illustrate the main results.

Necessary and sufficient conditions for optimal controls in viscous flow problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500028444 ◽

1994 ◽

Vol 124 (2) ◽

pp. 211-251 ◽

Cited By ~ 34

Author(s):

H. O. Fattorini ◽

S. S. Sritharan

Keyword(s):

Maximum Principle ◽

Viscous Flow ◽

Sufficient Conditions ◽

Optimal Controls ◽

The Maximum Principle ◽

Flow Problems ◽

Hamilton Jacobi Bellman Equation ◽

Hamilton Jacobi Bellman ◽

The Pontryagin Maximum Principle

A class of optimal control problems in viscous flow is studied. Main results are the Pontryagin maximum principle and the verification theorem for the Hamilton–Jacobi–Bellman equation characterising the feedback problem. The maximum principle is established by two quite different methods.

Controllability of nonlinear stochastic fractional higher order dynamical systems

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2019-0056 ◽

2019 ◽

Vol 22 (4) ◽

pp. 1063-1085

Author(s):

R. Mabel Lizzy ◽

K. Balachandran ◽

Yong-Ki Ma

Keyword(s):

Dynamical Systems ◽

Sufficient Conditions ◽

Banach Fixed Point Theorem ◽

Necessary And Sufficient Condition ◽

Bounded Operators ◽

Semilinear Systems ◽

Operator Functions ◽

Fractional System ◽

The Given

Abstract This paper deals with the study of controllability of stochastic fractional dynamical systems with 1 < α ≤ 2. Necessary and sufficient condition for controllability of linear stochastic fractional system is obtained. Sufficient conditions for controllability of stochastic fractional semilinear systems, integrodifferential systems, systems with neutral term, systems with delays in control and systems with Lévy noise is formulated and established. The solution is obtained in terms of Mittag-Leffler operator functions by considering bounded operators. The Banach fixed point theorem is used to obtain the desired results from an equivalent nonlinear integral equation of the given system.

THE CLASSICAL INCOMPRESSIBLE NAVIER-STOKES LIMIT OF THE BOLTZMANN EQUATION

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202591000137 ◽

1991 ◽

Vol 01 (02) ◽

pp. 235-257 ◽

Cited By ~ 74

Author(s):

CLAUDE BARDOS ◽

SEIJI UKAI

Keyword(s):

Boltzmann Equation ◽

Strong Solutions ◽

Navier Stokes ◽

Phase Method ◽

Stationary Phase Method ◽

Navier Stokes Equation ◽

Sharp Estimates ◽

The Cauchy Problem ◽

Global Strong Solutions ◽

The convergence hypothesis of Bardos, Golse, and Levermore,1 which leads to the incompressible Navier-Stokes equation as the limit of the scaled Boltzmann equation, is substantiated for the Cauchy problem with initial data small but independent of the Knudsen number ε. The uniform (in ε) existence of global strong solutions and their strong convergence as ε→0 are proved. A necessary and sufficient condition for the uniform convergence up to t=0, which implies the absence of the initial layer, is also established. The proof relies on sharp estimates of the linearized operators, which are obtained by the spectral analysis and the stationary phase method.

On stabilisation of solutions of the Cauchy problem for parabolic equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500019478 ◽

1976 ◽

Vol 76 (1) ◽

pp. 43-53 ◽

Cited By ~ 16

Author(s):

S. Kamin (Kamenomostskaya)

Keyword(s):

Cauchy Problem ◽

Parabolic Equations ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

The Cauchy Problem ◽

Image Position ◽

SynopsisThe author considers the solution of the Cauchy problem for an equationgiving necessary and sufficient conditions for the existence of

A two-parameter eigenvalue problem involving complex potentials

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500031449 ◽

1990 ◽

Vol 116 (1-2) ◽

pp. 177-191

Author(s):

M. Faierman

Keyword(s):

Hilbert Space ◽

Eigenvalue Problem ◽

Differential Equations ◽

Complex Potentials ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Parameter System ◽

Complete Set ◽

Two Parameter

SynopsisWe consider a two-parameter system of ordinary differential equations of the second order involving complex potentials and show that, unlike the case of real potentials, the eigenfunctions of the system do not necessarily form a complete set in the usual Hilbert space associated with the problem. We also give a necessary and sufficient condition for the eigenfunctions to be complete. Finally, we establish some results concerning the eigenvalues of the system.

On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions: Global well-posedness and decay property

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891616500168 ◽

2016 ◽

Vol 13 (03) ◽

pp. 633-659 ◽

Cited By ~ 7

Author(s):

Evgeny Yu. Panov

Keyword(s):

Cauchy Problem ◽

Conservation Laws ◽

Almost Periodic Functions ◽

Entropy Solutions ◽

Almost Periodic ◽

Periodic Functions ◽

Scalar Conservation Law ◽

The Cauchy Problem ◽

Scalar Conservation ◽

We study the Cauchy problem for a multidimensional scalar conservation law with merely continuous flux vector in the class of Besicovitch almost periodic functions. The existence and uniqueness of entropy solutions are established. We also uncover the necessary and sufficient condition for the decay of almost periodic entropy solutions as the time variable [Formula: see text]. Our results are then interpreted in the framework of conservation laws on the Bohr compact.

Existence conditions for two-parameter eigenvalue problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500012592 ◽

1981 ◽

Vol 91 (1-2) ◽

pp. 15-30 ◽

Cited By ~ 7

Author(s):

Paul Binding ◽

Patrick J. Browne ◽

Lawrence Turyn

Keyword(s):

Hilbert Spaces ◽

Eigenvalue Problems ◽

Sufficient Conditions ◽

Compact Resolvent ◽

Existence Conditions ◽

Image Position ◽

Adjoint Operators ◽

Bounded Below ◽

Two Parameter

SynopsisWe discuss necessary and sufficient conditions for the existence of eigentuples λ=(λl,λ2) and eigenvectors x1≠0, x2≠0 for the problem Wr(λ)xr = 0, Wr(λ)≧0, (*), where Wr(λ)= Tr + λ1Vr2, r=1,2. Here Tr and Vrs are self-adjoint operators on separable Hilbert spaces Hr. We assume the Vrs to be bounded and the Tr bounded below with compact resolvent. Most of our conditions involve the conesWe obtain results under various conditions on the Tr, but the following is typical:THEOREM. If (*) has a solution for all choices ofT1, T2then (a)0∉ V1UV2,(b)V1∩(—V2) =∅ and (c) V1⊂V2∪{0}, V2⊈V1∪{0}. Conversely, if (a) and (b) hold andV1⊈V2∪∩{0}, V2⊈ then (*) has a solution for all choices ofT1, T2.

On the Solutions of Some Linear Complex Quaternionic Equations

The Scientific World JOURNAL ◽

10.1155/2014/563181 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Cennet Bolat ◽

Ahmet İpek

Keyword(s):

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Sylvester Equation ◽

Real Matrix ◽

Matrix Representations ◽

Linear Complex ◽

The Real ◽

Special Cases ◽

Some complex quaternionic equations in the typeAX-XB=Care investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.