Uniqueness Of Optimal Control Research Articles

Optimal control for a vibrating string with variable axial load and damping gain

Abstract We consider an optimal control problem for a vibrating string with multiplicative distributive control. Multiplicative control is the coefficient u ( x, t ) of velocity term y t . Using multiplicative control, we bring the state solution to the desired profile under a quadratic cost of control.We prove the existence of optimal control. Criterion for characterization of the unique optimal control is provided in the form of optimality system.

Maximum principles for backward doubly stochastic systems with jumps and applications

We investigate the optimality conditions for stochastic control problems of backward doubly stochastic systems with jumps. Necessary and sufficient optimality conditions, where the control domains are convex and the control is allowed to enter into all the coefficients, are proved. The results are stated in the local form of stochastic maximum principle, as well as in the global form. Applying the stochastic maximum principles to doubly stochastic linear quadratic control problems, we obtain the unique optimal control. An example is given to illustrate the theoretical results.

Spatial dynamics and convergence: The spatial AK model

We study the optimal dynamics of an AK economy where population is uniformly distributed along the unit circle. Locations only differ in initial capital endowments. Spatio-temporal capital dynamics are described by a parabolic partial differential equation. The application of the maximum principle leads to necessary but non-sufficient first-order conditions. Thanks to the linearity of the production technology and the special spatial setting considered, the value function of the problem is found explicitly, and the (unique) optimal control is identified in feedback form. Despite constant returns to capital, we prove that the spatio-temporal dynamics, induced by the willingness of the planner to give the same (detrended) consumption over space and time, lead to convergence in the level of capital across locations in the long-run.

On the optimal harvesting of size-structured population dynamics

This work is concerned with a kind of optimal control problem for a size-structured biological population model. Well-posedness of the state system and an adjoint system are proved by means of Banach’s fixed point theorem. Existence and uniqueness of optimal control are shown by functional analytical approach. Optimality conditions describing the optimal strategy are established via tangent and normal cones technique. The results are of the first ones for this novel structure.

Capacitary Measures for Completely Monotone Kernels via Singular Control

We give a singular control approach to the problem of minimizing an energy functional for measures with given total mass on a compact real interval when energy is defined in terms of a completely monotone kernel. This problem occurs both in potential theory and when looking for optimal financial order execution strategies under transient price impact. In our setup, measures or order execution strategies are interpreted as singular controls, and the capacitary measure is the unique optimal control. The minimal energy, or equivalently the capacity of the underlying interval, is characterized by means of a nonstandard infinite-dimensional Riccati differential equation, which is analyzed in some detail. We then show that the capacitary measure has two Dirac components at the endpoints of the interval and a continuous Lebesgue density in between. This density can be obtained as the solution of a certain Volterra integral equation of the second kind.

Capacitary Measures for Completely Monotone Kernels Via Singular Control

We give a singular control approach to the problem of minimizing an energy functional for measures with given total mass on a compact real interval, when energy is defined in terms of a completely monotone kernel. This problem occurs both in potential theory and when looking for optimal financial order execution strategies under transient price impact. In our setup, measures or order execution strategies are interpreted as singular controls, and the capacitary measure is the unique optimal control. The minimal energy, or equivalently the capacity of the underlying interval, is characterized by means of a nonstandard infinite-dimensional Riccati differential equation, which is analyzed in some detail. We then show that the capacitary measure has two Dirac components at the endpoints of the interval and a continuous Lebesgue density in between. This density can be obtained as the solution of a certain Volterra integral equation of the second kind.

An optimal birth control problem for a dynamical population model with size-structure

This work is concerned with an optimal control problem for a size-structured population model, which takes fertility as the control variable. The existence and uniqueness of solutions to the basic state system and the dual system are proven via the Banach fixed point theorem. Necessary optimality conditions of first order are established in the form of an Euler–Lagrange system by the use of tangent–normal cone technique. The existence of a unique optimal controller is established by means of Ekeland’s variational principle. An example and some comments are presented.

No-weight design of H2 controllers for square plants

In this study, a reduced complexity design procedure for H 2 control problems of square plants is presented. First, all stabilising controllers are parameterised. Second, a modified inner-outer factorisation is defined for unstable plants and analytical formulas are developed. Third, the unique optimal controller is analytically derived by utilising the proposed parameterisation and the modified inner-outer factorisation. Finally, a simple tuning rule is developed for quantitative performance and robustness. The proposed procedure has three features: First, it is a no-weight design. The designer is not required to choose a weight. Second, this is an analytical design. The designer can directly use the developed design formulas and thus the design procedure is significantly simplified. Third, this is a quantitative design. The designer can design the controller for quantitative performance such as overshoot or stability margin. Numerical examples are given to illustrate the proposed method.

HJB Equations for the Optimal Control of Differential Equations with Delays and State Constraints, II: Verification and Optimal Feedbacks

This paper, which is the natural continuation of part I [S. Federico, B. Goldys, and F. Gozzi, SIAM J. Control Optim., 48 (2010), pp. 4910-4937], studies a class of optimal control problems with state constraints where the state equation is a differential equation with delays. In part I the problem is embedded in a suitable Hilbert space $H$ and the regularity of the associated Hamilton-Jacobi-Bellman equation is studied. The goal of the present paper is to exploit the regularity result of part I to prove a verification theorem and find optimal feedback controls for the problem. While it is easy to define a feedback control formally following the classical case, the proof of its existence and optimality is hard due to lack of full regularity of $V$ and to the infinite dimensionality of the problem. The theory developed is applied to study economic problems of optimal growth for nonlinear time-to-build models. In particular, we show the existence and uniqueness of optimal controls and their characterization as feedbacks.

Maximum Principle for Backward Doubly Stochastic Control Systems with Applications

We investigate the optimal control problems for backward doubly stochastic control systems. As a necessary condition of the optimal control we obtain a stochastic maximum principle. We found that the deduced stochastic Hamiltonian system exactly corresponds to a type of time-symmetric forward-backward stochastic differential equations, which was first introduced by Peng and Shi [C. R. Math. Acad. Sci. Paris, 336 (2003), pp. 773-778]. Applying the stochastic maximum principle to doubly stochastic linear quadratic problems, we obtain the unique optimal control. The existence and uniqueness of the solution is also obtained for a type of generalized forward-backward doubly stochastic differential equations derived from the maximum principle under some suitable assumptions. In particular, the result can be used in backward doubly stochastic linear quadratic optimal control problems. An example is given to illustrate the theoretical results.

Optimal boundary control with critical penalization for a PDE model of fluid–solid interactions

We study the finite-horizon optimal control problem with quadratic functionals for an established fluid-structure interaction model. The coupled PDE system under investigation comprises a parabolic (the fluid) and a hyperbolic (the solid) dynamics; the coupling occurs at the interface between the regions occupied by the fluid and the solid. We establish several trace regularity results for the fluid component of the system, which are then applied to show well-posedness of the Differential Riccati Equations arising in the optimization problem. This yields the feedback synthesis of the unique optimal control, under a very weak constraint on the observation operator; in particular, the present analysis allows general functionals, such as the integral of the natural energy of the physical system. Furthermore, this work confirms that the theory developed in Acquistapace et al. (Adv Diff Eq, [2])—crucially utilized here—encompasses widely differing PDE problems, from thermoelastic systems to models of acoustic-structure and, now, fluid-structure interactions.

Optimal scanning control of flexible structures in two-dimensional space

A class of optimal control problems for hyperbolic systems in two-dimensional space is considered. An approach is proposed to damp the undesirable vibrations in the structures by pointwise moving force actuators extending over the spatial region occupied by the structure. A class of performance indices is introduced that includes functions of the state variable, its first and second-order space derivatives and first-order time derivative evaluated at a preassigned terminal time, and a suitable penalty term involving the control forces. A maximum principle is given for such general scanning control problem that facilitates the determination of the unique optimal control. A solution method is developed for the active vibration control of plates of general shape. The implementation of the method is presented and the effectiveness of a single moving force actuator is investigated and compared to a single fixed force actuator by a specific numerical example.

Pollution's ambient problems and regularity of optimal cost function

We study pollution's ambient problems by using the optimal control theory applied to partial differential equations. We consider the problem to find the optimal way to eliminate pollution in the time, such that the concentration is close to a standard level which does not affect the ecological equilibrium when the source is pointwise. We consider a quadratic cost functional and we prove the existence and uniqueness of optimal control. We find a characterisation which makes possible the computing of optimal control. Additionally, we consider the problem moving the pointwise source. So, we define a function j(b) that associates to any point b ∈ Ω the optimal cost functional applied to the optimal control. We show that j is differentiable, provided that the controls are taken in a convenient subset of admissible functions satisfying the cone properties. We also find the point in Ω, for which the cost functional is minimum.

Optimal control of neutral systems with nonlinear base (The maximum principle perspective)

In this paper we examine the Euclidean controllability for a neutral system with a nonlinear base given by ddtD(t,xt) = f[(t,xt,u(t)] + B(t)u(t) * By a careful analysis of the maximum principle, necessary and sufficient conditions for the existence and uniqueness of optimal controls are deduced. This work is a great improvement of existing works providing a relationship between the attainable and reachable sets. JONAMP Vol. 11 2007: pp.269-274

Study of the optimal harvesting control for an elliptic problem with slow diffusion

In this paper our aim is to present a survey of known results of an optimal control problem with concave non-quadratic cost functional and a state equation arising from population dynamics. First, we study in detail the state equation, and then we show existence and uniqueness of optimal control and also a numerical approximation of the optimal control.

Forme standard de passage. Une alternative à la synthèse multi-objectif

This chapter presents a generalization of the Cross Standard Form (CSF) for a given plant and a given controller of arbitrary order (even, a two-degree of freedom compensator). The CSF is a canonical augmented standard plant whose H∞ or H2 unique optimal controller is a given controller, that is a solution to the general inverse optimal control problem. From the control design point of view, the general idea is to apply the CSF to a first controller in order to initialize an H∞ design procedure to handle complementary frequency-domain constraints or robustness specifications. An academic example and an industrial application in the field of aeronautics engineering are given.

Optimal Control of the Dynamic Viscoelastic Stress-Strain State of a Composite Body

New optimal control problems for the dynamic viscoelastic stress-strain state of a composite body with quadratic cost functionals are considered. Existence theorems for unique optimal controls are proved for all the cases considered.

Complex Optimal Control of a Thermally Stressed State of a Two-Component Body

New problems of optimal control for a thermally stressed state of a two-component solid body are considered in the paper. The body contains a thin and a weakly thermally penetrating inclusion. The existence of unique optimal controls is proved for every described case with quadratic cost functionals.

On Generic Existence and Uniqueness in Nonconvex Optimal Control Problems

We derive conditions for generic existence and uniqueness of optimal control and trajectories for some class of finite-dimensional optimal control problems in the absence of traditional convexity assumptions. It is shown that for these problems existence and uniqueness of optimal control for a given initial point x is equivalent to the differentiability of optimal value functions at x. These results are obtained in the general framework of representation formulas for (sub-)gradients of inf-envelope functions which appear in nonsmooth analysis.

Optimal birth control problems for nonlinear age-structured population dynamics

We study the least cost-size problem and the least cost-deviation problem for a nonlinear population model with age-dependence, which takes fertility rate as the control variable. The existence of a unique optimal control and the optimality conditions of first order are investigated by means of Ekeland's variational principle and normal cone technique. Our conclusion extends a known result in the literature.