Form Of Maximum Principle Research Articles

Maximum Principle for the Damped Boussinesq equation

In this paper, the optimal distributed control problem of an Damped Boussinesq equation is studied. Dubovitskii-Milyutin functional analytical approach is formulated for obtaining the necessary optimality condition in the form of Maximum principle for the system under consideration. Necessary optimality condition is presented in fixed final horizon case.

On partially observed optimal singular control of McKean–Vlasov stochastic systems: Maximum principle approach

In this paper, we study partially observed optimal stochastic singular control problems of general Mckean–Vlasov type with correlated noises between the system and the observation. The control variable has two components, the first being absolutely continuous and the second is a bounded variation, nondecreasing continuous on the right with left limits. The dynamic system is governed by Itô‐type controlled stochastic differential equation. The coefficients of the dynamic depend on the state process and of its probability law and the continuous control variable. In terms of a classical convex variational techniques, we establish a set of necessary conditions of optimal singular control in the form of maximum principle. Our main result is proved by applying Girsanov's theorem and the derivatives with respect to probability law in Lions' sense. To illustrate our theoretical result, we study partially observed linear‐quadratic singular control problem of McKean–Vlasov type.

Optimality conditions for a Timoshenko beam model

Optimality conditions for a Timoshenko beam model are derived in the maximum principle. For this aim, existence and uniqueness of the solution to the beam system and controllability properties of the system are discussed. Necessary optimality condition for the beam system, subjected to the rotary damping, external excitation, and mixed integral constraints including equality or/and inequality on the control function and state, is derived in the form of maximum principle. Besides obtaining that maximum principle is necessary condition for the optimality, it is also indicated sufficient optimality requirement is maximum principle in the presence of some convexity assumptions on the constraints.

Thermally induced vibration suppression in a thermoelastic beam structure

In this paper, the problem of thermally induced vibration suppression in a thermoelastic beam is studied. Physical equivalent of the present problem is that a thermoelastic beam is suddenly entering into daylight zone and vibrations are induced due to heating on the upper surface of the beam or thermoelastic beam in a spacecraft enters to intensive sunlight area just after leaving a shadow of a planet. Thermally induced vibrations are suppressed by means of minimum using of control forces to be applied to dynamic space actuators. Objective functional of the problem is chosen as a modified quadratical functional of the kinetic energy of the thermoelastic beam. Necessary optimality condition to be satisfied by an optimal control force is derived in the form of maximum principle, which converts the optimal vibration suppression problem to solving a system of distributed parameters system linked by initial-boundary-terminal conditions. Solution of the system is achieved via MATLAB? and simulated results reveal that thermally induced vibration suppression by means of dynamic space actuators are very effective and robust.

Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games

We study a kind of partial information non-zero sum differential games of mean-field backward doubly stochastic differential equations, in which the coefficient contains not only the state process but also its marginal distribution, and the cost functional is also of mean-field type. It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motions. We establish a necessary condition in the form of maximum principle and a verification theorem, which is a sufficient condition for Nash equilibrium point. We use the theoretical results to deal with a partial information linear-quadratic (LQ) game, and obtain the unique Nash equilibrium point for our LQ game problem by virtue of the unique solvability of mean-field forward-backward doubly stochastic differential equation.

A maximum principle related to volume growth and applications

In this paper, we derive a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold M for which there exists a bounded vector field X such that $$\langle \nabla f,X\rangle \ge 0$$ on M and $$\rm {div} X\ge af$$ outside a suitable compact subset of M, for some constant $$a>0$$ , under the assumption that M has either polynomial or exponential volume growth. We then use it to obtain some Bernstein-type results for hypersurfaces immersed into a Riemannian manifold endowed with a Killing vector field, as well as to some results on the existence and size of minimal submanifolds immersed into a Riemannian manifold endowed with a conformal vector field.

Nonzero-sum differential game of backward doubly stochastic systems with delay and applications

This paper is concerned with a kind of nonzero-sum differential game of backward doubly stochastic system with delay, in which the state dynamics follows a delayed backward doubly stochastic differential equation (SDE). To deal with the above game problem, it is natural to involve the adjoint equation, which is a kind of anticipated forward doubly SDE. We give the existence and uniqueness of solutions to delayed backward doubly SDE and anticipated forward doubly SDE. We establish a necessary condition in the form of maximum principle with Pontryagin's type for open-loop Nash equilibrium point of this type of game, and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a nonzero-sum differential game of linear-quadratic backward doubly stochastic system with delay.

Necessary Conditions for Optimality in Problems of Optimal Control of Systems with Discontinuous Right-Hand Side

We consider the problems of optimal control of a dynamical system whose right-hand side is discontinuous in the state variable and is linear in the control with sufficiently smooth coefficients in each of the half-spaces into which the space is divided by the switching hyperplane. The main attention is paid to the situation where there exist intervals on which the optimal trajectory lies on the switching surface. New nondegenerate necessary conditions for optimality are stated and proved in the maximum principle form. The obtained optimality conditions are compared with the already known conditions.

A maximum principle at infinity with applications to geometric vector fields

We derive a new form of maximum principle, applicable to a vector field with nonnegative divergence in a connected, oriented, complete noncompact Riemannian manifold. We then use it to obtain some applications to Killing vector fields. More precisely, we first show that, under a reasonable condition at infinity, an orientable, connected, complete noncompact hypersurface of a Riemannian manifold, transversal to a Killing vector field of constant norm and with nonnegative second fundamental form, is totally geodesic. We also deal with the case of a hypersurface of constant mean curvature – instead of nonnegative second fundamental form, and show that it has to be totally geodesic too.

On the total variation Wasserstein gradient flow and the TV-JKO scheme

We study the JKO scheme for the total variation, characterize the optimizers, prove some of their qualitative properties (in particular a form of maximum principle and in some cases, a minimum principle as well). Finally, we establish a convergence result as the time step goes to zero to a solution of a fourth-order nonlinear evolution equation, under the additional assumption that the density remains bounded away from zero, this lower bound is shown in dimension one and in the radially symmetric case.

Non-degenerate maximum principle for optimal control problems with discontinuous right-hand side

ABSTRACTIn the paper, optimal control problems with discontinuous right-hand side are considered. The main attention is paid to situations when there are intervals where optimal trajectory lies on a switching surface. It is shown that for such a situation the known optimality conditions degenerate: they are satisfied for any feasible control. Hence such conditions are not informative. New non-degenerate necessary optimality conditions in a form of Maximum principle are formulated and proved. Comparison of the obtained optimality conditions with ones known from the literature is presented.

Minimum Time Problem for <i>n×n</i> Co-operative Hyperbolic Lag Systems

In this paper, a minimum time problem for n×n co-operative hyperbolic systems involving Laplace operator and with time-delay is considered. First, the existence of a unique solution of such hyperbolic system with time-delay is proved. Then necessary conditions of a minimum time control are derived in the form of maximum principle. Finally the bang-bang principle and the approximate controllability conditions are investigated.

Variational principle for stochastic singular control of mean-field Lévy-forward-backward system driven by orthogonal Teugels martingales with application

We consider stochastic singular control for mean-field forward-backward stochastic differential equations, driven by orthogonal Teugels martingales associated with some Levy processes having moments of all orders and an independent Brownian motion. Under partial information, necessary and sufficient conditions for optimality in the form of maximum principle for this mean-field system are established by means of convex variation methods and duality techniques. As an illustration, this paper studies a partial information mean-variance portfolio selection problem driven by orthogonal Teugels martingales associated with gamma process as Levy process of bounded variation.

Variational principle for stochastic singular control of mean-field Lévy-forward-backward system driven by orthogonal Teugels martingales with application

We consider stochastic singular control for mean-field forward-backward stochastic differential equations, driven by orthogonal Teugels martingales associated with some Lévy processes having moments of all orders and an independent Brownian motion. Under partial information, necessary and sufficient conditions for optimality in the form of maximum principle for this mean-field system are established by means of convex variation methods and duality techniques. As an illustration, this paper studies a partial information mean-variance portfolio selection problem driven by orthogonal Teugels martingales associated with gamma process as Lévy process of bounded variation.

Production and process management: An optimal control approach

Optimal control and efficient management of industrial products are the key for sustainable development in industrial and process engineering. It is well-known that proper maintenance of process performance, ensuring the quality products after a long time operation of the system, is desirable in any industry. Nonlinear dynamical systems may play crucial role to appropriately design the model and obtain optimal control strategy in production and process management. This paper deals with a mathematical model in terms of ordinary differential equations (ODEs) that describe control of production and process arising in industrial engineering. The optimal control technique in the form of maximum principle, used to control the quality products in the operation processes, is applied to analyze the model. It is shown that the introduction of state constraint can be advantageous for obtaining good products during the longer operation process. We investigate the model numerically, using some known nonlinear optimal control solvers, and we present the simulation results to illustrate the significance of introducing state constraint onto the dynamics of the model.

A minimum-time control for Boolean control networks with impulsive disturbances

This paper investigates a Mayer-type optimal control and minimum-time control of a Boolean control network (BCN) with impulsive disturbances. Using the semi-tensor product, the BCN with impulsive disturbances is converted into algebraic discrete-time impulsive dynamic systems, and several necessary conditions for optimality are derived. Then we consider the problem of steering a BCN with impulsive disturbances from a given initial state to a desired state in minimal time. And a necessary condition, stated in the form of maximum principle, is obtained for a control to be time-optimal. It shows that the impulsive disturbances play an important role in the optimal control problem for BCNs. At last, a biological example is given to illustrate the effectiveness and advantage of the obtained results.

Mean-Field Maximum Principle for Optimal Control of Forward–Backward Stochastic Systems with Jumps and its Application to Mean-Variance Portfolio Problem

We study mean-field type optimal stochastic control problem for systems governed by mean-field controlled forward–backward stochastic differential equations with jump processes, in which the coefficients depend on the marginal law of the state process through its expected value. The control variable is allowed to enter both diffusion and jump coefficients. Moreover, the cost functional is also of mean-field type. Necessary conditions for optimal control for these systems in the form of maximum principle are established by means of convex perturbation techniques. As an application, time-inconsistent mean-variance portfolio selection mixed with a recursive utility functional optimization problem is discussed to illustrate the theoretical results.

Maximum principle for one kind of stochastic delay optimal control problem with state constraint

This paper is concerned with one kind of stochastic delay optimal control problem with state constraint, in which the control domain is convex and the control variable can enter the diffusion coefficient. Besides, the control domain is allowed to be unbounded. Necessary condition in the form of maximum principle is established. Sufficient optimality condition is also given, which can help to find optimal controls.

Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance

This paper is concerned with a maximum principle for a new class of non-zero sum stochastic differential games. Compared with the existing literature, the gamesystems in this paper are forward-backward systems in which the control variables consist of two components: the continuous controls and the impulse controls.Necessary optimality conditions and sufficient optimality conditions in the form of maximum principle are obtained respectively for open-loop Nash equilibriumpoint of the foregoing games. A fund management problem is used to shed light on the application of the theoretical results, and the optimal investmentportfolio and optimal impulse consumption strategy are obtained explicitly.

Nonzero Sum Differential Game of Mean-Field BSDEs with Jumps under Partial Information

This paper is concerned with a kind of nonzero sum differential game of mean-field backward stochastic differential equations with jump (MF-BSDEJ), in which the coefficient contains not only the state process but also its marginal distribution. Moreover, the cost functional is also of mean-field type. It is required that the control is adapted to a subfiltration of the filtration generated by the underlying Brownian motion and Poisson random measure. We establish a necessary condition in the form of maximum principle with Pontryagin’s type for open-loop Nash equilibrium point of this type of partial information game and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a partial information linear-quadratic (LQ) game.