Transversality condition and optimization of higher order ordinary differential inclusions

In the present paper, a Bolza problem of optimal control theory with a fixed time interval given by convex and nonconvex second-order differential inclusions (P_H) is studied. Our main goal is to derive sufficient optimality conditions for Cauchy problem of sth-order differential inclusions. The sufficient conditions including distinctive transversality condition are proved incorporating the Euler–Lagrange and Hamiltonian type inclusions. The basic concepts involved in obtaining optimality conditions are the locally adjoint mappings. Furthermore, the application of these results is demonstrated by solving the problems with third-order differential inclusions.     

On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems

For a class of infinite-horizon optimal control problems that appear in studies on economic growth processes, the properties of the adjoint variable in the relations of the Pontryagin maximum principle, defined by a formula similar to the Cauchy formula for the solutions to linear differential systems, are studied. It is shown that under a dominating discount type condition the adjoint variable defined in this way satisfies both the core relations of the maximum principle (the adjoint system and the maximum condition) in the normal form and the complementary stationarity condition for the Hamiltonian. Moreover, a new economic interpretation of the adjoint variable based on this formula is presented.     

Discrete Approximations and Optimization of Evolution Inclusions

The paper studies discrete approximations of nonconvex valued evolution inclusions with the right-hand side satisfying Kamke condition which is more general than the Lipschitz one and more convenient than the variant of the one-sided Lipschitz condition used in Donchev et al. (J Differ Equ 243:301–328, 2007). We extend an interesting previous result of Mordukhovich to a large class of evolution systems appearing in the theory of parabolic partial differential equations. Examples of control systems governed by partial differential equations are provided.     

Indefinite Boundary Eigenvalue Problems in a Pontrjagin Space Setting

We study eigenvalue problems Fy = λ Gy consisting of Hamiltonian systems of ordinary differential equations on a compact interval with symmetric λ-linear boundary conditions. The problems we are interested in are non-definite: neither left-nor right-definite. Instead of this, we give some weak condition on one coefficient of the Hamiltonian system which ensures that a hermitian form associated with the operator F has at most finitely many negative squares. This enables us to study the problem by the help of a compact self-adjoint operator in a Pontrjagin space and we obtain as a main result uniformly convergent eigenfunction expansions. In the final section, applications to formally self-adjoint differential equations of higher order are given.     

An Implicit Function Theorem for One-sided Lipschitz Mappings

Implicit function theorems are derived for nonlinear set valued equations that satisfy a relaxed one-sided Lipschitz condition. We discuss a local and a global version and study in detail the continuity properties of the implicit set-valued function. Applications are provided to the Crank–Nicolson scheme for differential inclusions and to the analysis of differential algebraic inclusions.     

Arrow-type sufficient optimality conditions for nondifferentiable optimal control problems with state constraints

In this paper a class of nondifferentiable optimal control problems governed by differential inclusions and subject to state variable inequality constraints is considered. Sufficient conditions using the concavity of the maximized Hamiltonian are given. Furthermore, a counterexample is presented that shows that in the nondifferentiable case the maximum principle does not form sufficient optimality conditions if the adjoint relation is formulated in terms of the ordinary Hamiltonian rather than the maximized one. Finally, it is shown that the sufficient conditions correspond to Clarke's necessary conditions with some additional assumptions such as concavity.     

Optimization of Second-Order Discrete Approximation Inclusions

The present article studies the approximation of the Bolza problem of optimal control theory with a fixed time interval given by convex and non-convex second-order differential inclusions (P _C ). Our main goal is to derive necessary and sufficient optimal conditions for a Cauchy problem of second-order discrete inclusions (P _D ). As a supplementary problem, discrete approximation problem (P _DA ) is considered. Necessary and sufficient conditions, including distinctive transversality, are proved by incorporating the Euler-Lagrange and Hamiltonian type of inclusions. The basic concept of obtaining optimal conditions is the locally adjoint mappings (LAM) and equivalence theorems, one of the most characteristic features of such approaches with the second-order differential inclusions that are peculiar to the presence of equivalence relations of LAMs. Furthermore, the application of these results are demonstrated by solving some non-convex problem with second-order discrete inclusions.     

与薛丁谔型谱问题相联系的孤子族的分解

The soliton hierarchy associated with a Schrodinger type spectral problem with four potentials is decomposed into a class of new finite-dimensional Hamiltonian systems by using the nonlinearized approach.It is worth to point that the solutions for the soliton hierarchy are reduced to solving the compatible Hamiltonian systems of ordinary differential equations.     

Necessary conditions for optimal control problems with state constraints

Necessary conditions of optimality are derived for optimal control problems with pathwise state constraints, in which the dynamic constraint is modelled as a differential inclusion. The novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid. An Euler Lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded, nonconvex sets and satisfies a mild `one-sided' Lipschitz continuity hypothesis. We recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by Ioffe. For problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one-sided Lipschitz hypothesis is replaced by a milder, local hypothesis. A recent `dualization' theorem permits us to infer a strengthened form of the Hamiltonian inclusion from the Euler Lagrange condition. The necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by Loewen and Rockafellar to achieve related necessary conditions for state constrained problems, and improve on available results in certain respects even when specialized to the state constraint free case.

Proofs make use of recent `decoupling' ideas of the authors, which reduce the optimization problem to one to which Pontryagin's maximum principle is applicable, and a refined penalization technique to deal with the dynamic constraint.

     

Limit circle invariance for two differential systems on time scales

In this paper we consider two linear differential systems on a time scale. Both systems depend linearly on a complex spectral parameter λ. We prove that if all solutions of these two systems are square integrable with respect to a given weight matrix for one value λ₀, then this property is preserved for all complex values λ. This result extends and improves the corresponding continuous time statement, which was derived by Walker (1975) for two non‐hermitian linear Hamiltonian systems, to appropriate differential systems on arbitrary time scales. The result is new even in the purely discrete case, or in the scalar time scale case, as well as when both time scale systems coincide. The latter case also generalizes a limit circle invariance criterion for symplectic systems on time scales, which was recently derived by the authors.     

On the method of interconnection and damping assignment passivity-based control for the stabilization of mechanical systems

Interconnection and damping assignment passivity-based control (IDA-PBC) is an excellent method to stabilize mechanical systems in the Hamiltonian formalism. In this paper, several improvements are made on the IDA-PBC method. The skew-symmetric interconnection submatrix in the conventional form of IDA-PBC is shown to have some redundancy for systems with the number of degrees of freedom greater than two, containing unnecessary components that do not contribute to the dynamics. To completely remove this redundancy, the use of quadratic gyroscopic forces is proposed in place of the skew-symmetric interconnection submatrix. Reduction of the number of matching partial differential equations in IDA-PBC and simplification of the structure of the matching partial differential equations are achieved by eliminating the gyroscopic force from the matching partial differential equations. In addition, easily verifiable criteria are provided for Lyapunov/exponential stabilizability by IDA-PBC for all linear controlled Hamiltonian systems with arbitrary degrees of underactuation and for all nonlinear controlled Hamiltonian systems with one degree of underactuation. A general design procedure for IDA-PBC is given and illustrated with examples. The duality of the new IDA-PBC method to the method of controlled Lagrangians is discussed. This paper renders the IDA-PBC method as powerful as the controlled Lagrangian method.     

Optimization of higher order differential inclusions with initial value problem

This paper concerns the sufficient conditions of optimality for initial value problem with higher order differential inclusions (HODIs) and free endpoint constraints. Formulation of the transversality conditions plays a substantial role in the next investigations without which hardly any necessary or sufficient conditions would be obtained. In terms of Euler–Lagrange and Hamiltonian forms the sufficient conditions of optimality both for convex and “non-convex” HODIs are based on the apparatus of locally adjoint mappings. Moreover, by applying the main result to a Bolza problem described by a polynomial differential operator with constant coefficients in terms of the adjoint differential operator the sufficient condition of optimality is obtained.     

On almost periodicity of delayed predator–prey model with mutual interference and discontinuous harvesting policies

The objective of this paper is to investigate the almost periodic dynamics for a class of delayed predator–prey model with mutual interference and Beddington–DeAngelis type functional response, in which the harvesting policies are modeled by discontinuous functions. Based on the theory of functional differential inclusions theory and set‐valued analysis, the solution in sense of Filippov of system with the discontinuous harvesting policies is given, and the local and global existence of positive the solution in sense of Filippov of the system is studied. By employing generalized differential inequalities, some useful Lemmas are obtained. After that, sufficient conditions which guarantee the permanence of the system are obtained in view of the constructed Lemmas. By constructing some suitable generalized Lyapunov functional, a series of useful criteria on existence, uniqueness, and global attractivity of the almost positive periodic solution to the system are derived in view of functional differential inclusions theory and nonsmooth analysis theory. Some suitable examples together with their numeric simulations are given to substantiate the theoretical results and to illustrate various dynamical behaviors of the system. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiplicity of solutions for second-order Hamiltonian systems with impulses

In this paper, we study the existence of infinitely many solutions for second-order Hamiltonian systems with impulses. By using an infinitely many critical points theorem and Fountain theorem, we obtain some new criteria for guaranteeing that the impulsive Hamiltonian systems have infinitely many solutions. No symmetric condition on the nonlinear term is assumed. Some examples are also given in this paper to illustrate our main results.     

Non–limit-circle and limit-point criteria for symplectic and linear Hamiltonian systems

Several necessary and/or sufficient conditions for the existence of a non–square-integrable solution of symplectic dynamic systems with general linear dependence on the spectral parameter on time scales are established and a sufficient condition for the limit-point case is derived. Almost all presented results are new even in the continuous and discrete cases, that is, for the linear Hamiltonian differential systems and for the discrete symplectic systems, respectively.     

On Systems of Boundary Value Problems for Differential Inclusions

Herein we consider the existence of solutions to second-order, two-point boundary value problems （BVPs） for systems of ordinary differential inclusions. Some new Bernstein-Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow of the treatment of systems of BVPs without growth restrictions.     

Conservation laws and Hamilton’s equations for systems with long-range interaction and memory

Using the fact that extremum of variation of generalized action can lead to the fractional dynamics in the case of systems with long-range interaction and long-term memory function, we consider two different applications of the action principle: generalized Noether’s theorem and Hamiltonian type equations. In the first case, we derive conservation laws in the form of continuity equations that consist of fractional time–space derivatives. Among applications of these results, we consider a chain of coupled oscillators with a power-wise memory function and power-wise interaction between oscillators. In the second case, we consider an example of fractional differential action 1-form and find the corresponding Hamiltonian type equations from the closed condition of the form.     

Periodic solutions of large norm of Hamiltonian systems

Hamiltonian systems of ordinary differential equations are studied. The only assumption made on the Hamiltonian is appropriately rapid growth at infinity. It is proved that for any given period, there is an unbounded sequence of periodic solutions of the system having the given period.     

Semilocal convergence analysis for the modified Newton-HSS method under the Hölder condition

The present paper is concerned with theoretical properties of the modified Newton-HSS method for large sparse non-Hermitian positive definite systems of nonlinear equations. Assuming that the nonlinear operator satisfies the Hölder continuity condition, a new semilocal convergence theorem for the modified Newton-HSS method is established. The Hölder continuity condition is milder than the usual Lipschitz condition. The semilocal convergence theorem is established by using the majorizing principle, which is based on the concept of majorizing sequence given by Kantorovich. Two real valued functions and two real sequences are used to establish the convergence criterion. Furthermore, a numerical example is given to show application of our theorem.     

Hamiltonian operators in differential algebras

We develop a previously proposed algebraic technique for a Hamiltonian approach to evolution systems of partial differential equations including constrained systems and propose a defining system of equations (suitable for computer calculations) characterizing the Hamiltonian operators of a given form. We demonstrate the technique with a simple example.