Order Necessary Conditions For Optimality Research Articles

Sparse optimal control problems with intermediate constraints: Necessary conditions

AbstractThis article treats optimal sparse control problems with multiple constraints defined at intermediate points of the time domain. For such problems with intermediate constraints, we first establish a new Pontryagin maximum principle that provides first order necessary conditions for optimality in such problems. Then we announce and employ a new numerical algorithm to arrive at, in a computationally tractable fashion, optimal state‐action trajectories from the necessary conditions given by our maximum principle. Several detailed illustrative examples are included.

Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations

<p style='text-indent:20px;'>The convective Brinkman-Forchheimer (CBF) equations describe the motion of incompressible viscous fluids through a rigid, homogeneous, isotropic, porous medium and is given by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_t{\boldsymbol{u}}-\mu \Delta{\boldsymbol{u}}+({\boldsymbol{u}}\cdot\nabla){\boldsymbol{u}}+\alpha{\boldsymbol{u}}+\beta|{\boldsymbol{u}}|^{r-1}{\boldsymbol{u}}+\nabla p = {\boldsymbol{f}},\ \nabla\cdot{\boldsymbol{u}} = 0. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>In this work, we consider some distributed optimal control problems like total energy minimization, minimization of enstrophy, etc governed by the two dimensional CBF equations with the absorption exponent <inline-formula><tex-math id="M1">\begin{document}$ r = 1,2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>. We show the existence of an optimal solution and the first order necessary conditions of optimality for such optimal control problems in terms of the Euler-Lagrange system. Furthermore, for the case <inline-formula><tex-math id="M3">\begin{document}$ r = 3 $\end{document}</tex-math></inline-formula>, we show the second order necessary and sufficient conditions of optimality. We also investigate an another control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is 2D CBF equations, using optimal control techniques.</p>

An efficient numerical solution for time switching optimal control problems

In this paper, an efficient computational algorithm for the solution of Hamiltonian boundary value problems arising from bang-bang optimal control problems is presented. For this purpose, at first, based on the Pontryagin’s minimum principle, the first order necessary conditions of optimality are derived. Then, an indirect shooting method with control parameterization, in which the control function is replaced with a piecewise constant function with values and switching points taken as unknown parameters, is presented. Thereby, the problem is converted to the solution of the shooting equation, in which the values of the control function and the switching points as well the initial values of the costate variables are unknown parameters. The important advantages of this method are that, the obtained solution satisfies the first order optimality conditions, further the switching points can be captured accurately which is led to an accurate solution of the bangbang problem. However, solving the shooting equation is nearly impossible without a very good initial guess. So, in order to cope with the difficulty of the initial guess, a homotopic approach is combined with the presented method. Consequently, no priori assumptions are made on the optimal control structure and number of the switching points, and sensitivity to the initial guess for the unknown parameters is resolved too. Illustrative examples are included at the end and efficiency of the method is reported.

Rate Constrained Discrete-time Maximum Principle

Abstract Actuators in real-world systems operate under limited bandwidth and saturation constraints. Mathematically, these limitations manifest as certain types of constraints being imposed on the sequence of control action. While saturation constraints have been studied extensively, bandwidth limitations in the actuators have received very little attention in the control community. While an attempt has been made in the direction of incorporating frequency constraints at the synthesis stage, rate constraints on the control at the design stage has not been studied extensively in the regime of discrete-time. This article tries to fill in that gap by incorporating rate constraints on the control trajectories. We propose a new discrete-time Pontryagin maximum principle (PMP) with rate constraints being imposed on the control and derive a set of first order necessary conditions for optimality. As an illustration, the linear quadratic (LQ) optimal control problem under rate constraints has been considered.

Bifurcation preserving discretisations of optimal control problems

The first order optimality conditions of optimal control problems (OCPs) can be regarded as boundary value problems for Hamiltonian systems. Variational or symplectic discretisation methods are classically known for their excellent long term behaviour. As boundary value problems are posed on intervals of fixed, moderate length, it is not immediately clear whether methods can profit from structure preservation in this context. When parameters are present, solutions can undergo bifurcations, for instance, two solutions can merge and annihilate one another as parameters are varied. We will show that generic bifurcations of an OCP are preserved under discretisation when the OCP is either directly discretised to a discrete OCP (direct method) or translated into a Hamiltonian boundary value problem using first order necessary conditions of optimality which is then solved using a symplectic integrator (indirect method). Moreover, certain bifurcations break when a non-symplectic scheme is used. The general phenomenon is illustrated on the example of a cut locus of an ellipsoid.

First order necessary conditions of optimality for the two dimensional tidal dynamics system

<p style='text-indent:20px;'>In this work, we consider the two dimensional tidal dynamics equations in a bounded domain and address some optimal control problems like total energy minimization, minimization of dissipation of energy of the flow, etc. We also examine an another interesting control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is the tidal dynamics, using optimal control techniques. For these cases, different distributed optimal control problems are formulated as the minimization of suitable cost functionals subject to the controlled two dimensional tidal dynamics system. The existence of an optimal control as well as the first order necessary conditions of optimality for such systems are established and the optimal control is characterized via the adjoint variable. We also establish the uniqueness of optimal control in small time interval.</p>

Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations

Abstract In this paper, we formulate a distributed optimal control problem related to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The distributed optimal control problem is framed as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We describe the first order necessary conditions of optimality via the Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.

Real-Time Optimization of Periodic Systems: A Modifier-Adaptation Approach

Abstract Modifier-Adaptation methodologies have been widely used to overcome plant-model mismatch and control a system to its steady-state optimal setpoint. They use gradient information of the real plant to design modifiers that correct the model, so that the first order necessary conditions for optimality of the model-based problem converge to those of the optimal one. In this paper, we get rid of the hypothesis that the plant optimum needs to be an equilibrium point. Instead, we only require it to be a periodic trajectory. We show the behaviour of the proposed approach by means of a motivating example that highlights the necessity of this formulation in cases where the system changes periodically through time.

Dynamic programming and feedback analysis of the two dimensional tidal dynamics system

In this work, we consider the controlled two dimensional tidal dynamics equations in bounded domains. A distributed optimal control problem is formulated as the minimization of a suitable cost functional subject to the controlled 2D tidal dynamics equations. The existence of an optimal control is shown and the dynamic programming method for the optimal control of 2D tidal dynamics system is also described. We show that the feedback control can be obtained from the solution of an infinite dimensional Hamilton-Jacobi equation. The non-differentiability and lack of smoothness of the value function forced us to use the method of viscosity solutions to obtain a solution of the infinite dimensional Hamilton-Jacobi equation. The Bellman principle of optimality for the value function is also obtained. We show that a viscosity solution to the Hamilton-Jacobi equation can be used to derive the Pontryagin maximum principle, which give us the first order necessary conditions of optimality. Finally, we characterize the optimal control using the adjoint variable.

Save the pine forests of wilt disease using a fractional optimal control strategy

Abstract There is no doubt that the pine trees contribute greatly to the strengthening of the economy and the domestic wealth in many countries, where they produce dense grain, maintain the soil, especially in the sand slopes, uses wood in fuel and is one of the most important sources of recreation. However, over the past few years, pine forests have been plagued by many diseases, notably the Pine Wilt Disease (PWD), which poses a major threat to these forests. So, in this article, we investigate the best strategy to reduce and eliminate this disease using fractional optimal control strategy. Here, we introduce a mathematical system of equations contains the fractional order derivative with respect to time which describes the transmission dynamics of PWD. We calculate the general basic reproduction number R 0 and discuss the stability of a disease-free and endemic equilibrium in the proposed model. Furthermore, a Fractional Optimal Control Problem (FOCP) with three proposed controls is formulated, and the fractional order necessary conditions of optimality by using the Ponntryagin maximum principle are derived. We apply both analytical and numerical techniques on the FOCP with suggested controls to demonstrate the effective control strategies to prevent the transmission of the PWD. The numerical simulation of the suggested FOCP is presented and these results are expected to be helpful in the early treatment of PWD-infected trees.

Discrete time optimal control with frequency constraints for non-smooth systems

We present a Pontryagin maximum principle for discrete time optimal control problems with (a) pointwise constraints on the control actions and the states, (b) smooth frequency constraints on the control and the state trajectories, and (c) nonsmooth dynamical systems. Pointwise constraints on the states and the control actions represent desired and/or physical limitations on the states and the control values; such constraints are important and are widely present in the optimal control literature. Constraints of type (b), while less standard in the literature, effectively serve the purpose of describing important properties of inertial actuators and systems. The conjunction of constraints of type (a) and (b) is a relatively new phenomenon in optimal control but is important for the synthesis control trajectories with a high degree of fidelity. The maximum principle established here provides first order necessary conditions for optimality that serve as a starting point for the synthesis of control trajectories corresponding to a large class of constrained motion planning problems that have high accuracy in a computationally tractable fashion. Moreover, the ability to handle a reasonably large class of nonsmooth dynamical systems that arise in practice ensures broad applicability of our theory.

Least-squares solution of a class of optimal space guidance problems via Theory of Connections

In this paper, we apply a newly developed method to solve boundary value problems for differential equations to solve optimal space guidance problems in a fast and accurate fashion. The method relies on the least-squares solution of differential equations via orthogonal polynomials expansion and constrained expression as derived via Theory of Connection (ToC). The application of the optimal control theory to derive the first order necessary conditions for optimality, yields a Two-Point Boundary Value Problem (TPBVP) that must be solved to find state and costate. Combining orthogonal polynomials expansion and ToC, we solve the TPBVP for a class of optimal guidance problems including energy-optimal landing on planetary bodies and fixed-time optimal intercept for a target-interceptor scenario. The performance analysis in terms of accuracy shows the potential of the proposed methodology as applied to optimal guidance problems.

Solar sail trajectory optimization of multi-asteroid rendezvous mission

An interplanetary trajectory optimization of the multi-asteroid rendezvous is investigated for a solar sail-based spacecraft. The optimal control problem is formulated using the calculus of variations and Pontryagin's maximum principle and the first order necessary conditions for optimality of the problem are derived. To achieve the optimal solution, the indirect method transforms the optimal control problem into a multiple point boundary value problem and solves it via shooting method, and the pseudospectral method discretizes the state and control variables and solves the NLP problem. In this paper, a numerical method combining the global optimization algorithm and a nonlinear equation solver is proposed to get the optimal solution of the indirect method. Additionally, costates are estimated to validate the optimality of the solution obtained via the pseudospectral method. Finally, numerical examples are given to verify the effectiveness of the proposed methods. The results indicate that the optimal solutions to the problems with two and three targets can be solved using the proposed indirect method algorithm. As the number of visiting targets is more than three, the direct method will be more efficient.

A discrete-time Pontryagin maximum principle on matrix Lie groups

In this article we derive a Pontryagin maximum principle (PMP) for discrete-time optimal control problems on matrix Lie groups. The PMP provides first order necessary conditions for optimality; these necessary conditions typically yield two point boundary value problems, and these boundary value problems can then be solved to extract optimal control trajectories. Constrained optimal control problems for mechanical systems, in general, can only be solved numerically, and this motivates the need to derive discrete-time models that are accurate and preserve the non-flat manifold structures of the underlying continuous-time controlled systems. The PMPs for discrete-time systems evolving on Euclidean spaces are not readily applicable to discrete-time models evolving on non-flat manifolds. In this article we bridge this gap and establish a discrete-time PMP on matrix Lie groups. Our discrete-time models are derived via discrete mechanics, (a structure preserving discretization scheme) leading to the preservation of the underlying manifold under the dynamics, thereby resulting in greater numerical accuracy of our technique. This PMP caters to a class of constrained optimal control problems that includes point-wise state and control action constraints, and encompasses a large class of control problems that arise in various field of engineering and the applied sciences.

A Boundary Control Problem for the Viscous Cahn–Hilliard Equation with Dynamic Boundary Conditions

A boundary control problem for the viscous Cahn–Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.

Global Minimization of an Infinite Collection of Instances of the Total Annualized Cost Problem for Compressor Sequences

In this work, a novel hybrid method is proposed for the global solution of an infinite collection of instances of the minimum total annualized cost (TAC) problem for a series of gas compressors and coolers that brings a gas from a given pressure and temperature to a specified final pressure and the same temperature. The gas is considered to possess a constant compressibility factor, and a constant, ideal gas, constant pressure, heat capacity. The collection of TAC problem instances considered is parametrized by the overall compression ratio. The proposed method combines the TAC problem’s first and second order necessary conditions of optimality with interval analysis, to determine converging upper and lower bounds to the globally optimal solution and associated optimal solution vector components, for all considered instances of the TAC problem. Two case studies are presented to illustrate the novel solution method and the impact of economies of scale on the global optimum.

Morse Families In Optimal Control Problems

We geometrically describe optimal control problems in terms of Morse families in the Hamiltonian framework. These geometric structures allow us to recover the classical first order necessary conditions for optimality and the starting point to run an integrability algorithm. Moreover the integrability algorithm is adapted to optimal control problems in such a way that the trajectories originated by discontinuous controls are also obtained. From the Hamiltonian viewpoint we obtain the equations of motion for optimal control problems in the Lagrangian formalism by means of a proper Lagrangian submanifold. Singular optimal control problems and overdetermined ones are also studied in the paper.

On a Finite Branch and Bound Algorithm for the Global Minimization of a Concave Power Law Over a Polytope

In this paper, a finite branch-and-bound algorithm is developed for the minimization of a concave power law over a polytope. Linear terms are also included in the objective function. Using the first order necessary conditions of optimality, the optimization problem is transformed into an equivalent problem consisting of a linear objective function, a set of linear constraints, a set of convex constraints, and a set of bilinear complementary constraints. The transformed problem is then solved using a finite branch-and-bound algorithm that solves two convex problems at each of its nodes. The method is illustrated by means of an example from the literature.

Second order necessary conditions of optimality for stochastic systems with variable delay

Second order necessary conditions of optimality for stochastic systems with variable delay

A variational method using fractional order Hilbert spaces for tomographic reconstruction of blurred and noised binary images

We provide in this article a refined functional analysis of the Radon operator restricted to axisymmetric functions, and show that it enjoys strong regularity properties in fractional order Hilbert spaces. This study is motivated by a problem of tomographic reconstruction of binary axially symmetric objects, for which we have available one single blurred and noised snapshot. We propose a variational approach to handle this problem, consisting in solving a minimization problem settled in adapted fractional order Hilbert spaces. We show the existence of solutions, and then derive first order necessary conditions for optimality in the form of optimality systems.