Study of optimal control problems on the half-line by the averaging method

We justify the application of the averaging method to optimal control problems for systems of differential equations on the half-line. For optimal control problems for systems of differential equations linear in the control, we prove the existence of optimal controls for the exact and averaged problems. We show that an optimal control in the averaged problem is ɛ-optimal in the exact problem.     

On the numerical solution of certain boundary control problems for vibrating media in one space-dimension

This paper is concerned with the numerical solution of control problems which consist of minimizing certain quadratic functionals depending on control functions in L ₂[0,1] for some given time T > 0 and bounded with respect to the maximum norm. These control functions act upon the boundary conditions of a vibrating system in one space-dimension which is governed by a wave equation of spatial order 2n They are to be chosen in such a way that a given initial state of vibration at time zero is transferred into the state of rest. This requirement can be expressed by an infinite system of moment equations to be satisfied by the control functions

The control problem is approximated by replacing this infinite system by finitely many, say N, equations (truncation) and by choosing piecewise constant functions as controls (discretization). The resulting problem is a quadratic optimization problem which is solved very efficiently by a multiplier method

Convergence of the solutions of the approximating problems to the solution of the control problem, as N tends to infinity and the discretization is infinitely refined, is shown under mild assumptions. Numerical results are presented for a vibrating beam     

Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

In this paper, an effective numerical approach based on a new two‐dimensional hybrid of parabolic and block‐pulse functions (2D‐PBPFs) is presented for solving nonlinear partial quadratic integro‐differential equations of fractional order. Our approach is based on 2D‐PBPFs operational matrix method together with the fractional integral operator, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro‐differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h³) . The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier.     

A posteriori error estimates of hp spectral element methods for optimal control problems with L2-norm state constraint

In this paper, we investigate a distributed optimal control problem governed by elliptic partial differential equations with L²-norm constraint on the state variable. Firstly, the control problem is approximated by hp spectral element methods, which combines the advantages of the finite element methods with spectral methods; then, the optimality conditions of continuous system and discrete system are presented, respectively. Next, hp a posteriori error estimates are derived for the coupled state and control approximation. In the end, a projection gradient iterative algorithm is given, which solves the optimal control problems efficiently. Numerical experiments are carried out to confirm that the numerical results are in good agreement with the theoretical results.

     

Optimal synchronization of two different in‐commensurate fractional‐order chaotic systems with fractional cost function

This article investigates the optimal synchronization of two different fractional‐order chaotic systems with two kinds of cost function. We use calculus of variations for minimizing cost function subject to synchronization error dynamics. We introduce optimal control problem to solve fractional Euler–Lagrange equations. Optimal control signal and minimum time of synchronization are obtained by proposed method. Examples show the optimal synchronization of two different systems with two different cost functions. First, we use an ordinary integer cost function then we use a fractional‐order cost function and comparing the results. Finally, we suggest a cost function which has the optimal solution of this problem, and we can extend this solution to solve other synchronization problems. © 2016 Wiley Periodicals, Inc. Complexity 21: 401–416, 2016     

Su alcuni problemi fortemente ben posti di controllo ottimo

Summary Let P_n be a sequence of optimal control problems with fixed end times (described by ordinary differential equations, with pointwise and norm (of controls) constraints, and with general functional). Let P₀ be an ? unperturbed ? problem (of the same type). In this paper theorems are obtained about the strong convergence (in some L^p) of optimal controls of P_n to some optimal control of P₀, about the uniform convergence x_n → x₀ of states, and the property thatmin P_n min P₀. As corollaries, convergence theorems for some calculus of variations problems can be derived. Weak convergence theorems of optimal controls of P_n to an optimal one of P₀ were considered in[7]. A general abstract theorem about strong convergence of minimum points, generalizing a result in[5], is proved.

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Indirizzo dell'autore: Istituto di Matematica, via L. B. Alberti 4 - 16132 Genova. Lavoro eseguito nell'ambito del Centro di Matematica e di Fisica Teorica del C.N.R. presso l'Università di Genova.

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Entrata in Redazione il 4 maggio 1972.     

A class of degenerate fractional evolution systems in banach spaces

By using the notion of strongly (B, p)-sectorial operator and fractional differential calculus, we analyze the unique solvability of the Cauchy and Showalter problems for a class of degenerate fractional evolution systems. The results are used for the analysis of partial differential equations of fractional order with respect to the time variable.     

Optimal Solutions to Relaxation in Multiple Control Problems of Sobolev Type with Nonlocal Nonlinear Fractional Differential Equations

We introduce the optimality question to the relaxation in multiple control problems described by Sobolev-type nonlinear fractional differential equations with nonlocal control conditions in Banach spaces. Moreover, we consider the minimization problem of multi-integral functionals, with integrands that are not convex in the controls, of control systems with mixed nonconvex constraints on the controls. We prove, under appropriate conditions, that the relaxation problem admits optimal solutions. Furthermore, we show that those optimal solutions are in fact limits of minimizing sequences of systems with respect to the trajectory, multicontrols, and the functional in suitable topologies.     

Optimal Solutions of Fractional Nonlinear Impulsive Neutral Stochastic Functional Integro-Differential Equations

Abstract

In this article, we consider a new class of fractional impulsive neutral stochastic functional integro-differential equations with infinite delay in Hilbert spaces. First, by using stochastic analysis, fractional calculus, analytic α-resolvent operator and suitable fixed point theorems, we prove the existence of mild solutions and optimal mild solutions for these equations. Second, the existence of optimal pairs of system governed by fractional impulsive partial stochastic integro-differential equations is also presented. The results are obtained under weaker conditions in the sense of the fractional power arguments. Finally, an example is given for demonstration.     

Regular Degenerate Separable Differential Operators and Applications

The boundary value problems for differential-operator equations with variable coefficients, degenerated on all boundary are studied. Several conditions for the separability, fredholmness and resolvent estimates in L _p -spaces are given. In applications degenerate Cauchy problem for parabolic equation, boundary value problems for degenerate partial differential equations and systems of degenerate elliptic equations on cylindrical domain are studied.     

Esistenza e condizioni necessarie per controlli ottimi di sistemi definiti da equazioni integrali

Summary In the first two sections of this paper existence theorems are proved for optimal control problems described either by Fredholm systems of integral equations (with control appearing nonlinearly) on a given (possibly infinite) measure space, eventually in presence of eigenvalues, or by Urysohn systems, both with pointwise and integral constraints on controls and states. Controllability is assumed throughout the paper. In the last section necessary conditions are proved for optimal controls of Fredholm systems (a minimum principle in pointwise, or ? differential ? form), and theorems (that seem to be new also for the corresponding ordinary differential control systems) are deduced about bang-bang and regularization properties of the optimal control (s) (which turns out to be a continuous function except for a finite number of points, if positivity and semicontinuity assumptions hold about the data): in some particular cases the optimal control will be piecewise constant. All these results are independent on the known ones.

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Lavoro eseguito nell'ambito del Centro di Matematica e Fisica Teorica del C.N.R. presso l'Università di Genova.

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Entrata in Redazione il 6 maggio 1971.     

p-Moment Stability of Stochastic Nonlinear Delay Systems with Impulsive Jump and Markovian Switching

Abstract

This article is concerned with the problem of p-moment stability of stochastic differential delay equations with impulsive jump and Markovian switching. In this model, the features of stochastic systems, delay systems, impulsive systems, and Markovian switching are all taken into account, which is scarce in the literature. Based on Lyapunov–Krasovskii functional method and stochastic analysis theory, we obtain new criteria ensuring p-moment stability of trivial solution of a class of impulsive stochastic differential delay equations with Markovian switching.     

The minimax approach for a class of variable order fractional differential equation

This paper introduces an approximate solution for Liouville‐Caputo variable order fractional differential equations with order 0 < α(t) ≤ 1 . The solution is adapted using a family of fractional‐order Chebyshev functions with unknown coefficients. These coefficients have been obtained by using an optimization approach based on minimax technique and the least pth optimization function. Several linear and nonlinear fractional‐order differential equations are discussed using the proposed technique for fixed and variable order fractional‐order derivatives. Moreover, the response of RC charging circuit with variable order fractional capacitor is studied for different cases. Several comparisons with related published techniques have been added to illustrate the accuracy of the proposed approach.     

Riccati equations for generalized singular estimate control systems

We consider the Bolza problem associated with boundary/point control systems governed by strongly continuous semigroups. In continuation of our work in Lasiecka and Tuffaha [I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by C ₀–semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008), pp. 229–246; I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control systems, Control Cybernet 38(4B) (2009), pp. 1429–1460], we yet extend the theory to a more general class of control problems that are not analytic providing sharp blow-up rates for the regularity. Solvability of the associated Riccati equations and an optimal feedback synthesis are established. The presence of unbounded control actions, such as boundary/point controls, naturally lead to a singularity at the terminal point t?=?T of the optimal control and of the corresponding feedback operator as before. The class of control systems considered in this article is a generalization to the class usually referred to in the literature as ‘Singular Estimate Control Systems’. The prototype is still that of a PDE system consisting of coupled hyperbolic parabolic dynamics interacting on an interface with point/boundary control. The distinct feature of the class considered in this article is that the degree of unboundedness in the control is stronger than that allowed in the usual singular estimate control system configuration, giving rise to less regular optimal state trajectories.     

The control of pseudohyperbolic systems

Certain problems concerning the control of processes described by pseudohyperbolic-type equations are studied. Inequalities in negative norms are obtained. These inequalities imply the existence and uniqueness of the generalized solution of a boundary-value problem with right-hand sides belonging to a negative space, which makes it possible to consider the optimal problem in the class of generalized effects. The existence of the optimal control is proved. Some examples of pulse optimal control problems are given. Bibliography:5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 80, 1996, pp. 68–77.     

Approximate Controllability of Semilinear Fractional Stochastic Dynamic Systems with Nonlocal Conditions in Hilbert Spaces

A class of dynamic control systems described by semilinear fractional stochastic differential equations of order 1 < q < 2 with nonlocal conditions in Hilbert spaces is considered. Using solution operator theory, fractional calculations, fixed-point technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for nonlocal approximate controllability of semilinear fractional stochastic dynamic systems is formulated and proved by assuming the associated linear system is approximately controllable. As a remark, the conditions for the exact controllability results are obtained. Finally, an example is provided to illustrate the obtained theory.     

Derived Sets for Weak Multiobjective Optimization Problems with State and Control Variables

The concept of a K-gradient, introduced in Ref. 1 in order to generalize the concept of a derived convex cone defined by Hestenes, is extended to weak multiobjective optimization problems including not only a state variable, but also a control variable. The new concept is employed to state multiplier rules for the local solutions of such dynamic multiobjective optimization problems. An application of these multiplier rules to the local solutions of an abstract multiobjective optimal control problem yields general necessary optimality conditions that can be used to derive concrete maximum principles for multiobjective optimal control problems, e.g., problems described by integral equations with additional functional constraints.     

带有p-Laplace算子的分数阶边值问题的特征区间

本文研究了一类带有p-Laplace算子的分数阶微分方程两点边值问题.利用锥上的不动点定理,得到了这类边值问题的特征区间,推广了整数阶边值问题情形的结论.     

Some Solvable Stochastic Control Problems With Delay

We consider optimal harvesting of systems described by stochastic differential equations with delay. We focus on those situations where the value function of the harvesting problem depends on the initial path of the process in a simple way, namely through its value at 0 and through some weighted averages

A verification theorem of variational inequality type is proved. This is applied to solve explicitly some classes of optimal harvesting delay problems     

Strong controllability and optimal control of the heat equation with a thermal source

In this paper we consider an optimal control system described byn-dimensional heat equation with a thermal source. Thus problem is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective function. Here we assume there is no constraints on control. This problem is reduced to a moment problem.We modify the moment problem into one consisting of the minimization of a positive linear functional over a set of Radon measures and we show that there is an optimal measure corresponding to the optimal control. The above optimal measure approximated by a finite combination of atomic measures. This construction gives rise to a finite dimensional linear programming problem, where its solution can be used to determine the optimal combination of atomic measures. Then by using the solution of the above linear programming problem we find a piecewise-constant optimal control function which is an approximate control for the original optimal control problem. Finally we obtain piecewise-constant optimal control for two examples of heat equations with a thermal source in one-dimensional.