First-order Necessary Conditions For Optimality Research Articles

Optimal control of the Fokker-Planck equation under state constraints in the Wasserstein space

We analyze a problem of optimal control of the Fokker-Planck equation with state constraints in the Wasserstein space of probability measures. We give first-order necessary conditions for optimality in the form of a mean-field game system of partial differential equations associated with an exclusion condition. Under suitable geometric conditions on the constraint we prove that optimal controls are Lipschitz continuous.

Coherent Quantum LQG Controllers with Luenberger Dynamics

This paper is concerned with the coherent quantum linear-quadratic-Gaussian control problem of minimising an infinite-horizon mean square cost for a measurement-free field-mediated interconnection of a quantum plant with a stabilising quantum controller. The plant and the controller are multimode open quantum harmonic oscillators, governed by linear quantum stochastic differential equations and coupled to each other and the external multichannel bosonic fields in the vacuum state. We discuss an interplay between the quantum physical realizability conditions and the Luenberger structure associated with the classical separation principle. This leads to a quadratic constraint on the controller gain matrices, which is formulated in the framework of a swapping transformation for the conjugate positions and momenta in the canonical representation of the controller variables. For the class of coherent quantum controllers with the Luenberger dynamics, we obtain first-order necessary conditions of optimality in the form of algebraic equations, involving a matrix-valued Lagrange multiplier.

Globally Optimal SISO H2-Norm Model Reduction Using Walsh’s Theorem

We present a novel methodology for single-input single-output (SISO) H2-norm model reduction that guarantees global optimality of the obtained solution(s). By exploiting Walsh’s theorem, which is an elegant formulation of the first-order necessary conditions for optimality, we reformulate the model reduction problem as a multiparameter eigenvalue problem (MEVP), the real-valued eigentuples of which characterize the globally optimal solution(s) of the model reduction problem. While aiming for global optimality comes at the cost of a combinatorial growth of the problem complexity for increasing model orders, the novel methodology allows us to tackle larger problems compared to the few other globally optimal approaches in the literature. In particular, the degree of the obtained MEVP is independent of the order of the original higher order and obtained reduced-order model, a property that is favorable from a computational point of view. We perform three numerical experiments to illustrate the effectiveness of the methodology.

Optimal control of a nonconserved phase field model of Caginalp type with thermal memory and double obstacle potential

In this paper, we investigate optimal control problems for a nonlinear state system which constitutes a version of the Caginalp phase field system modeling nonisothermal phase transitions with a nonconserved order parameter that takes thermal memory into account. The state system, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. It consists of two nonlinearly coupled partial differential equations that govern the phase dynamics and the universal balance law for internal energy, written in terms of the phase variable and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. We extend recent results obtained for optimal control problems in which the free energy governing the phase transition was differentiable (i.e., of regular or logarithmic type) to the nonsmooth case of a double obstacle potential. As is well known, in this nondifferentiable case standard methods to establish the existence of appropriate Lagrange multipliers fail. This difficulty is overcome utilizing of the so-called deep quench approach. Namely, the double obstacle potential is approximated by a family of (differentiable) logarithmic ones for which the existence of optimal controls and first-order necessary conditions of optimality in terms of the adjoint state variables and a variational inequality are known. By proving appropriate bounds for the adjoint states of the approximating systems, we can pass to the limit in the corresponding first-order necessary conditions, thereby establishing meaningful first-order necessary optimality conditions also for the case of the double obstacle potential.

Direct coupling coherent quantum observers with discounted mean square performance criteria and penalized back-action

This paper is concerned with quantum harmonic oscillators consisting of a quantum plant and a directly coupled coherent quantum observer. We employ discounted quadratic performance criteria in the form of exponentially weighted time averages of the second-order moments of the system variables. Small-gain-theorem bounds are obtained for the back-action of the observer on the covariance dynamics of the plant in terms of the plant–observer coupling. A coherent quantum filtering (CQF) problem is formulated as the minimization of the discounted mean square of an estimation error, with which the dynamic variables of the observer approximate those of the plant. The cost functional also involves a quadratic penalty on the plant–observer coupling matrix in order to mitigate the back-action effect. For the discounted mean square optimal CQF problem with penalized back-action, we establish the first-order necessary conditions of optimality in the form of algebraic matrix equations. By using the Hamiltonian structure of the Heisenberg dynamics and Lie-algebraic techniques, this set of equations is represented in a more explicit form for equally dimensioned plant and observer. For a class of such observers with autonomous estimation error dynamics, we obtain a solution of the CQF problem and outline a homotopy method. The computation of the performance criteria and the observer synthesis are illustrated by numerical examples.

Optimal Control Problems with Sparsity for Tumor Growth Models Involving Variational Inequalities

This paper treats a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type. The evolution of the tumor fraction is governed by a variational inequality corresponding to a double obstacle nonlinearity occurring in the associated potential. In addition, the control and state variables are nonlinearly coupled and, furthermore, the cost functional contains a nondifferentiable term like the L^1-norm in order to include sparsity effects which is of utmost relevance, especially time sparsity, in the context of cancer therapies as applying a control to the system reflects in exposing the patient to an intensive medical treatment. To cope with the difficulties originating from the variational inequality in the state system, we employ the so-called deep quench approximation in which the convex part of the double obstacle potential is approximated by logarithmic functions. For such functions, first-order necessary conditions of optimality can be established by invoking recent results. We use these results to derive corresponding optimality conditions also for the double obstacle case, by deducing a variational inequality in terms of the associated adjoint state variables. The resulting variational inequality can be exploited to also obtain sparsity results for the optimal controls.

Generalized Regularization of Constrained Optimal Control Problems

Constraints in optimal control problems introduce challenges with traditional indirect methods. Bang-bang/singular solutions with discontinuous or indefinite control laws add further difficulty in numerical solution. Recent efforts in control regularization strategies have sought to overcome these limitations. Regularization generates a smoothed constraint transformation of a multiphase Hamiltonian boundary value problem to a single-phase unconstrained problem. This work develops a new approach to regularization using orthogonal error-control saturation functions. The method is developed for problems in bang-bang/singular form. The method is then applied to problems of general Hamiltonian structure using system extension and differential control. Applications in state constraint regularization are discussed. A key feature of the new approach is to eliminate ambiguity of the control law derived from the first-order necessary conditions of optimality. Results show desirable stability and convergence in numerical continuation. The method is applied to classical problems in optimal control, as well as problems of interest in aerospace mission design.

Data-Driven Approximation of the Perron-Frobenius Operator Using the Wasserstein Metric

This manuscript introduces a regression-type formulation for approximating the Perron-Frobenius Operator by relying on distributional snapshots of data. These snapshots may represent densities of particles. The Wasserstein metric is leveraged to define a suitable functional optimization in the space of distributions. The formulation allows seeking suitable dynamics so as to interpolate the distributional flow in function space. A first-order necessary condition for optimality is derived and utilized to construct a gradient flow approximating algorithm. The framework is exemplified with numerical simulations.

Необходимые условия оптимальности в одной дискретной граничной задаче управления динамикой популяции

One discrete problem of optimal population dynamics control with a controllable initial condition is considered. The process is described by a nonlinear system of Fredholm-type difference equations. An analogue of the L.S. Pontryagin’ maximum principle is proved by imposing a series of smoothness conditions on the right side the equation. The linearized maximum principle and the analogue of the Euler equation are proved for the case of convexity and openness of the control domain, which are first-order necessary conditions for optimality.

First-order necessary conditions of optimality for the optimal control of two-dimensional convective Brinkman–Forchheimer equations with state constraints

This work is concerned about some distributed optimal control problems governed by two-dimensional convective Brinkman–Forchheimer (CBF) equations, which describe the motion of incompressible viscous fluid through a rigid, homogeneous, isotropic, porous medium. We address a distributed optimal control problem of minimizing a suitable cost functional subjected to 2D CBF equations with state constraints. We establish the existence of an optimal solution and obtain the first-order necessary conditions of optimality for such optimal control problems.

Analysis and Optimal Velocity Control of a Stochastic Convective Cahn\u2013Hilliard Equation

A Cahn–Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for the randomness appearing at the microscopic level both in the phase-separation itself and in the flow-inducing process. The call for a random component in the convection term stems naturally from applications, as the fluid’s stirring procedure is usually caused by mechanical or magnetic devices. Well-posedness of the state system is addressed, and optimisation of a standard tracking type cost with respect to the velocity control is then studied. Existence of optimal controls is proved, and the Gâteaux–Fréchet differentiability of the control-to-state map is shown. Lastly, the corresponding adjoint backward problem is analysed, and the first-order necessary conditions for optimality are derived in terms of a variational inequality involving the intrinsic adjoint variables.

A composite framework for co-optimization of spacecraft trajectory and propulsion system

Indirect optimization methods convert optimal control problems (OCPs) into two- or multi-point boundary-value problems. A highly desirable feature of indirect methods, specifically for space applications, is that high-resolution trajectories can be generated, which satisfy the first-order necessary conditions of optimality. A recently developed Composite Smoothing Control (CSC) framework is utilized to formulate and solve the problem of simultaneous trajectory optimization and propulsion sub-system design of spacecraft. A reasonable parameterized breakdown of the spacecraft mass is adopted, which captures the impact of power produced by the solar arrays and its contribution to the total spacecraft mass. Thus, the implicit trade-offs can be considered in the indirect optimization approach. The function space co-optimization problem of spacecraft power subsystem parameters along with the main trajectory is solved with the objective to maximize the payload delivered. The proposed framework amounts to an invariant embedding that reduces the original, difficult-to-solve, multi-point boundary-value problem into a two-point boundary-value problem with continuous, differentiable control inputs. Utility of the proposed construct is demonstrated through a low-thrust, multi-revolution, multi-year rendezvous maneuver to asteroid Dionysus with a variable-specific-impulse, variable-thrust modeled engine. This is the first time that indirect optimization methods have tackled such a complex co-optimization problem using the CSC framework.

Optimal bilinear control of a reparable multi-state system

This work addresses an optimal bilinear control design representing the corrective maintenance for a reparable multi-state system. The primary interest is to optimise the availability of the system, which is defined as the probability that the system is operating properly when it is requested for use. The system model considered in our current work is governed by coupled transport and integro-differential equations. A corrective maintenance policy is represented by the repair rate, which depends on the distributed repair time. The objective is to determine an optimal repair rate that maximises the availability of the system in good mode over a given system running period. This essentially leads to a bilinear control problem set in a nonreflexive Banach space using -optimisation. A rigorous proof of existence of an optimal controller and the first-order necessary conditions of optimality are addressed. Numerical experiments are conducted to demonstrate the theoretical results.

A robust pseudospectral method for numerical solution of nonlinear optimal control problems

In the present paper, a robust pseudospectral method for efficient numerical solution of nonlinear optimal control problems is presented. In the proposed method, at first, based on the Pontryagin's minimum principle, the first-order necessary conditions of optimality which are led to the Hamiltonian boundary value problem are derived. Then, utilizing a pseudospectral method for discretization, the nonlinear optimal control problem is converted to a system of nonlinear algebraic equations. However, the need to have a good initial guess may lead to a challenging problem for solving the obtained system of nonlinear equations. So, an optimization approach is introduced to simplify the need of a good initial guess. Numerical findings of some benchmark examples are presented at the end and computational features of the proposed method are reported.

Optimal Multiplexing of Discrete-Time Constrained Control Systems on Matrix Lie Groups

In this article, we study a constrained optimal control problem for an ensemble of control systems in a centralized setting. Each system evolves on a matrix Lie group, and must satisfy given state and control action constraints pointwise in time. In addition, the controller must be shared between the plants in the sense that at any time instant the control signal may be sent to only one plant while minimizing a given objective function. We provide first-order necessary conditions for optimality in the form of a Pontryagin maximum principle for such optimal control problems.

Maximum Principle for Some Optimal Control Problems Governed by 2D Nonlocal Cahn–Hillard–Navier–Stokes Equations

This work concerns some optimal control problems associated with the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The Cahn–Hilliard–Navier–Stokes model consists of a Navier–Stokes equation governing the fluid velocity field coupled with a convective Cahn–Hilliard equation for the relative concentration of one of the fluids. A distributed optimal control problem is formulated as the minimization of a cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We establish the first-order necessary conditions of optimality by proving the Pontryagin maximum principle for optimal control of such system via the seminal Ekeland variational principle. The optimal control is characterized using the adjoint variable. We also study an another optimal control problem of finding the unknown optimal initial data. Optimal data initialization problem is also known as the data assimilation problems in meteorology, where determining the correct initial condition is very crucial for the future predictions.

A modified pseudospectral method for indirect solving a class of switching optimal control problems

In the present paper, an efficient pseudospectral method for solving the Hamiltonian boundary value problems arising from a class of switching optimal control problems is presented. For this purpose, based on the Pontryagin's minimum principle, the first-order necessary conditions of optimality are derived. Then, by partitioning the time interval related to the problem under study into some subintervals, the states (and costates) and control functions are approximated on each subintervals with piecewise interpolating polynomials based on Legendre–Gauss–Radau points and a piecewise constant function, respectively. As a result, solution of the problem is turned into solution of a number of algebraic equations, in which the values of the states (and costates) and control functions at Legendre–Gauss–Radau points as well as switching and terminal points are allowed to be unknown. Numerical examples are presented at the end to show the efficiency of the proposed method.

Measurement-based Feedback Control of Linear Quantum Stochastic Systems with Quadratic-Exponential Criteria

This paper is concerned with a risk-sensitive optimal control problem for a feedback connection of a quantum plant with a measurement-based classical controller. The plant is a multimode open quantum harmonic oscillator driven by a multichannel quantum Wiener process, and the controller is a linear time invariant system governed by a stochastic differential equation. The control objective is to stabilize the closed-loop system and minimize the infinite-horizon asymptotic growth rate of a quadratic-exponential functional (QEF) which penalizes the plant variables and the controller output. We combine a frequency-domain representation of the QEF growth rate, obtained recently, with variational techniques and establish first-order necessary conditions of optimality for the state-space matrices of the controller.

Practical Homotopy Methods for Finding the Best Minimum-Fuel Transfer in the Circular Restricted Three-Body Problem

Finding a solution to a low-thrust minimum-fuel transfer trajectory optimization problem in the circular restricted three-body problem (CRTBP) is already known to be extremely difficult due to the strong nonlinearity and the discontinuous bang-bang control structure. What is less commonly known is that such a problem can have many local solutions that satisfy all the first-order necessary conditions of optimality. In this paper, the existence of multiple solutions in the CRTBP is fully demonstrated, and a practical technique that involves multiple homotopy procedures is proposed, which can robustly search as many as desired local solutions to determine the best solution with the optimal performance index. Due to the common failure of the general homotopy method in tracking a homotopy path, the parameter bounding fixed-point (PBFB) homotopy is coupled to construct the double-homotopy, which can overcome the difficulties by tracking the discontinuous homotopy path. Furthermore, the PBFB homotopy is capable to achieve multiple solutions that lie on separate homotopy path branches. The minimum-time problem is first solved by the double-homotopy on the thrust magnitude to infer a reasonable transfer time. Then the auxiliary problem is solved by the double-homotopy and the PBFP homotopy, and multiple local solutions with smooth control profiles are obtained. By tracking homotopy paths starting from these solution points, multiple solutions to the minimum-fuel problem are obtained, and the best solution is eventually identified. Numerical demonstration of the minimum-fuel transfers from a GEO to an Earth-Moon $L_{1}$ halo orbit is presented to substantiate the effectiveness of the technique.

Boundary control problem and optimality conditions for the Cahn–Hilliard equation with dynamic boundary conditions

This paper is concerned with a boundary control problem for the Cahn–Hilliard equation coupled with dynamic boundary conditions. In order to handle the control problem, we restrict our analysis to the case of regular potentials defined on the whole real line, assuming the boundary potential to be dominant. The existence of optimal control, the Fréchet differentiability of the control-to-state operator between appropriate Banach spaces, and the first-order necessary conditions for optimality are addressed. In particular, the necessary condition for optimality is characterised by a variational inequality involving the adjoint variables.