Adjoint System Research Articles

Discrete Carleman estimates and application to controllability for a fully-discrete parabolic operator with dynamic boundary conditions

We consider a fully-discrete approximation of a 1-D heat equation with dynamic boundary conditions for which we provide a controllability result. The proof of this result is based on a relaxed observability inequality for the corresponding adjoint system. This is conducted using a suitable Carleman estimate for such models where the discrete parameters h and Δt are connected to one of the large Carleman parameters.

CFD-based geometrical shape optimization of a packed-bed reactor combining multi-objective and adjoint system methods

This paper presents the development of a geometric shape optimization methodology based on the so-called ”Hadamard boundary variation” method for performing very general domain deformations, and the related concept of domain differentiation. The resulting method is used to determine the optimal configuration of a two-dimensional packed-bed reactor that simultaneously optimizes its conversion rate and fluid energy dissipation, and where a homogeneous first-order reaction or a catalytic surface reaction takes place. The considered multi-objective optimization problem is subjected to four constraints: the process model constraints consisting of the Navier–Stokes, continuity and mass balance equations, an iso-volume and two manufacturing constraints. The approach to solve the problem is based on the linear scalarization method which converts the multi-objective problem into a single objective problem. The adjoint system method is used to compute the gradient of the performance indices and constraints. Since the indices are conflicting, the solution of the problem is a set of solutions, called Pareto front. Each optimal solution is evaluated using multi-attribute utility theory (MAUT) to determine the best optimal shape of the reactor. Finally, the resulting shape is fabricated using a 3D printing technique and will be experimentally validated.

Optimal Control and Parameters Identification for the Cahn–Hilliard Equations Modeling Tumor Growth

This paper is dedicated to the setting and analysis of an optimal control problem for a two-phase system composed of two non-linearly coupled Chan–Hilliard-type equations. The model describes the evolution of a tumor cell fraction and a nutrient-rich extracellular water volume fraction. The main objective of this paper is the identification of the system’s physical parameters, such as the viscosities and the proliferation rate, in addition to the controllability of the system’s unknowns. For this purpose, we introduce an adequate cost function to be optimized by analyzing a linearized system, deriving the adjoint system, and defining the optimality condition. Eventually, we provide a numerical simulation example illustrating the theoretical results. Finally, numerical simulations of a tumor growing in two and three dimensions are carried out in order to illustrate the evolution of such a clinical situation and to possibly suggest different treatment strategies.

Uniqueness and stability for inverse source problem for fractional diffusion-wave equations

Abstract This paper is devoted to the inverse problem of determining the spatially dependent source in a time fractional diffusion-wave equation, with the aid of extra measurement data at a subboundary. A uniqueness result is obtained by using the analyticity and the newly established unique continuation principle provided that the coefficients are all temporally independent. We also derive a Lipschitz stability of our inverse source problem under a suitable topology whose norm is given via the adjoint system of the fractional diffusion-wave equation.

Observability of a 1D Schrödinger Equation with Time-Varying Boundaries

We discuss the observability of a one-dimensional Schrödinger equation on certain time-dependent domain. In linear moving case, we give the exact boundary and pointwise internal observability for arbitrary time. For the general moving, we provide exact boundary observability when the curve satisfies some certain conditions. By duality theory, we establish the controllability of adjoint system.

Finite strain, laminate stress minimization with Newton iteration and time integration

We minimize failure criteria with respect to element-wise fiber orientation in laminae (and laminates) undergoing finite strains. A fully mechanical optimization approach is adopted: the analysis is encapsulated in a Newmark time integration equivalent to Nesterov’s first-order minimization algorithm, with Newton iteration followed by the solution of an adjoint system to obtain analytical sensitivities. This is implemented in our in-house software, Simplas. We consider transversely isotropic elasticity; and hyperelasticity, via the homogenized model of an incompressible Neo-Hookean material filled with cylindrical (fiber-like) pores. Two stress-based criteria are adopted: Tsai–Wu and modified Tsai–Hill. A finite strain solid-shell element, known to be locking-free, is used and here extended to perform the sensitivity operations. Three examples of optimized transversely isotropic elastic composites are shown, exhibiting remarkable advantages when compared with traditional optimization algorithms. One example is dedicated to finding optimal cylindrical void orientation in a hyperelasticity framework. The algorithm works for very high values of deformation. During stress minimization, stiffness can either decrease or increase, and this was observed in the numerical experiments.

An Effective Algorithm for the Stability and Bifurcation in a DDE Model of Gene Expression

The stability and Hopf bifurcation of gene expression models with a mechanism of delayed state feedback are considered. An effective algorithm for the calculations on the delay stable interval of the equilibrium point, the direction, and stability of the bifurcating periodic solution is also proposed. The τ-decomposition strategy is applied to tackle the issue of local stability, and the explicit formula for the delay stable interval is provided. In addition, the asymptotical behaviors of the bifurcation solutions are investigated by the center manifold theorem and normal form theory. The direction and stability of the Hopf bifurcation are determined naturally. In addition, a subtle bilinear form of the adjoint system is proposed to calculate the bifurcation parameters directly. Finally, the correctness and effectiveness of our results and algorithm are verified by typical numerical examples.

The null boundary controllability for the Mullins equation with periodic boundary conditions

In this paper, we study the null controllability of the Mullins equation with the control acting on the periodic boundary. Firstly, using the duality relation between controllability and observability, we express the controllability condition in terms of the solution of the backward adjoint system. After showing the existence and uniqueness of the solution of the adjoint system, we determine the admissible initial data class since the system is not always controllable under these boundary conditions. Finally, using this spectral analysis, we reduce the null controllability problem to the moment problem and solve the problem on this admissible initial class.

A Deterministic Adjoint-Based Semi-Analytical Algorithm for Fast Response Change Computations in Proton Therapy

In this paper we propose a solution to the need for a fast particle transport algorithm in Online Adaptive Proton Therapy capable of cheaply, but accurately computing the changes in patient dose metrics as a result of changes in the system parameters. We obtain the proton phase-space density through the product of the numerical solution to the one-dimensional Fokker-Planck equation and the analytical solution to the Fermi-Eyges equation. Moreover, a corresponding adjoint system was derived and solved for the adjoint flux. The proton phase-space density together with the adjoint flux and the metric (chosen as the energy deposited by the beam in a variable region of interest) allowed assessing the accuracy of our algorithm to different perturbation ranges in the system parameters and regions of interest. The algorithm achieved negligible errors () for small Hounsfield unit (HU) perturbation ranges (–40 HU to 40 HU) and small to moderate errors (3% to 17%) – in line with the well-known limitation of adjoint approaches – for large perturbation ranges (–400 HU to 400 HU) in the case of most clinical interest where the region of interest surrounds the Bragg peak. Given these results coupled with the capability of further improving the timing performance it can be concluded that our algorithm presents a viable solution for the specific purpose of Online Adaptive Proton Therapy.

Optimal control and stability analysis of an age-structured SEIRV model with imperfect vaccination.

Vaccination programs are crucial for reducing the prevalence of infectious diseases and ultimately eradicating them. A new age-structured SEIRV (S-Susceptible, E-Exposed, I-Infected, R-Recovered, V-Vaccinated) model with imperfect vaccination is proposed. After formulating our model, we show the existence and uniqueness of the solution using semigroup of operators. For stability analysis, we obtain a threshold parameter $ R_0 $. Through rigorous analysis, we show that if $ R_0 < 1 $, then the disease-free equilibrium point is stable. The optimal control strategy is also discussed, with the vaccination rate as the control variable. We derive the optimality conditions, and the form of the optimal control is obtained using the adjoint system and sensitivity equations. We also prove the uniqueness of the optimal controller. To visually illustrate our theoretical results, we also solve the model numerically.

First order necessary condition for stochastic evolution control systems with random generators

&lt;p style='text-indent:20px;'&gt;The main purpose of this paper is to establish the first order necessary optimality condition for optimal control problems of stochastic evolution systems with random generators. For our controlled system, both the drift and the diffusion terms contain the control variable, and the control region is a nonempty closed subset of a separable Hilbert space. Compared with the existing works, the main difficulties here are to prove the well-posedness of the control and the adjoint systems. We obtain the well-posedness of the adjoint system in the sense of mild solutions and that of the control system by means of the stochastic transposition method. In addition, the variational analysis approach is employed to handle the nonconvexity of the control region when deriving our optimality condition.&lt;/p&gt;

A hierarchical age-structured model of optimal vermin management by contraception.

Taking the reproduction law of vermin into consideration, we formulate a hierarchical age-structured model to describe the optimal management of vermin by contraception control. It is shown that the model is well-posed, and the solution has a separable form. The existence of optimal management policy is established via a minimizing sequence and the use of compactness, while the structure of optimal strategy is obtained by using an adjoint system and normal cones. To show the compactness, we use the Fréchet-Kolmogorov theorem and its generalization. To construct the adjoint system, we give some continuity results. Finally, an illustrative example is given.

Boundary null controllability for the heat equation with dynamic boundary conditions

&lt;p style='text-indent:20px;'&gt;In this paper, we are concerned with the boundary controllability of heat equation with dynamic boundary conditions. More precisely, we prove that the equation is null controllable at any positive time by means of a boundary control supported on an arbitrary subboundary. The proof of the main result combines a new boundary Carleman estimate and some regularity estimates for the adjoint system, with an explicit dependence with respect to the final time. This technique allows us to overcome a new difficulty that arises when absorbing a normal derivative term.&lt;/p&gt;

Analysis of diffusive size-structured population model and optimal birth control

&lt;p style='text-indent:20px;'&gt;This work addresses the optimal birth control problem for invasive species in a spatial environment. We apply the method of semigroups to qualitatively analyze a size-structured population model in which individuals occupy a position in a spatial environment. With insect population in mind, we study the optimal control problem which takes fertility rate as a control variable. With the help of adjoint system, we derive optimality conditions. We obtain the optimality conditions by fixing the birth rate on three different sets. Using Ekeland's variational principle, the existence, and uniqueness of optimal birth controller to the given population model which minimizes a given cost functional is shown. A concrete example is also given to see the behaviour of population density. Outcomes of our article are new and complement the existing ones.&lt;/p&gt;

Принцип максимума Понтрягина и непрямой метод спуска в задаче оптимального импульсного управления нелокальным уравнением переноса

We study a singular problem of optimal control of a nonlocal transport equation in the space of probability measures, in which the structure of the drivng vector field with respect to the control variable is somewhat equivalent to the affine one, while the set of controls is norm-unbounded and constrained in the integral sense only. We show that the problem at hand admits an impulse-trajectory relaxation in terms of discontinuous time reparameterization. This relaxation provides a correct statement of the variational problem in the class of control inputs constrained in both pointwise and integral senses. For the relaxed problem, we derive a new form of the Pontryagin’s maximum principle (PMP) with a separate adjoint system of linear balance laws on the space of signed measures. In contrast to the canonical formulation of the PMP in terms of the Hamiltonian equation on the cotangent bundle of the state space, our form allows one to formulate an indirect descent method for optimal impulsive control analogous to classical gradient descent. We expose a version of this method, namely, an algorithm of the steepest descent with an internal line search of the Lagrange multiplier associated with the integral bound on control. The algorithm is proven to monotonically converge to a PMP-extremal up to a subsequence.

Controllability results for cascade systems of m coupled N-dimensional stokes and Navier-stokes systems by N – 1 scalar controls

In this paper we deal with the controllability properties of a system of m coupled Stokes systems or m coupled Navier-Stokes systems. We show the null-controllability of such systems in the case where the coupling is in a cascade form and when the control acts only on one of the systems. Moreover, we impose that this control has a vanishing component so that we control a m × N state (corresponding to the velocities of the fluids) by N — 1 distributed scalar controls. The proof of the controllability of the coupled Stokes systems is based on a Carleman estimate for the adjoint system. The local null-controllability of the coupled Navier-Stokes systems is then obtained by means of the source term method and a Banach fixed point.

Optimal borel measure-valued controls to the viscous Cahn–Hilliard–Oberbeck–Boussinesq phase-field system on two-dimensional bounded domains

We consider an optimal control problem for the two-dimensional viscous Cahn-Hilliard-Ober beck-Boussinesq system with controls that take values in the space of regular Borel measures. The state equation models the interaction between two incompressible non-isothermal viscous fluids. Local distributed controls with constraints are applied in either of the equations governing the dynamics for the concentration, mean velocity, and temperature. Necessary and sufficient conditions characterizing local optimality in terms of the Lagrangian will be demonstrated. These conditions will be obtained through regularity results for the associated adjoint system, a priori estimates for the solutions of the linearized system in a weaker norm compared to that of the state space, and the Lebesgue decomposition of Borel measures.

Adjoint system in the shooting method to solve boundary value problems

The shooting method is used to solve a boundary value problem with separated and explicit constraints. To obtain approximations of an unknown initial values there are considered arguments based on the adjoint differential system attached to the given differential system. Finally the Newton-Kantorovich iterations are regained.

USE OF ANALYTICAL MODEL FOR SYNTHESIS OF ALGORITHMS FOR CONTROL OF TRANSPORT CONVEYOR PARAMETERS

This study presents a methodology for synthesizing optimal control algorithms for the flow parameters of a conveyor-type transport system with a variable transport delay. A multi-section transport conveyor is a complex dynamic system with a variable transport delay. The transport conveyor is an important element of the production system, used to synchronize technological operations and move material. The Analytical PiKh-model of the conveyor section was used as a model for designing a control system for flow parameters. The characteristic dimensionless parameters of the conveyor section are introduced and the similarity criteria for the conveyor sections are determined. The model of a conveyor section in a dimensionless form is used to develop a methodology for synthesizing algorithms for optimal control of the flow parameters of a transport conveyor section. The dependencies between the value of the input and output material flow of the section are determined, taking into account the initial distribution of the material along the conveyor section, variable transport delay, restrictions on the specific density of the material, and restrictions on the speed of the belt. The dependencies between the value of the input and output material flow for the case of a constant transport delay are analyzed. A technique for synthesizing algorithms for optimal belt speed control based on the PiKh-model of a conveyor section is presented. As a simplification, a two-stage belt speed control is considered. Particular attention is paid to the methodology for synthesizing optimal control algorithms based on the energy management methodology (TOU-Tariffs). The criteria of control quality are introduced and problems of optimal control of flow parameters of the transport system are formulated. Taking into account differential connections and restrictions on phase variables and admissible controls, which are typical for the conveyor section, the Pontryagin function and the adjoint system of equations are written. As examples demonstrating the design of optimal control, algorithms for optimal control of the flow parameters of the transport system are synthesized and analysis of optimal controls is performed.

Calculus of variations for estimation in ODE–PDE landslide-like models with discrete-time asynchronous measurements

Motivated by some landslide models, and related estimation challenges, this paper presents an optimal estimation method for state and parameter in a special class of so-called ODE–PDE system based on the adjoint method for discrete-time asynchronous measurements. This system is described by a pair of coupled Ordinary Differential Equation (ODE) and Partial Differential Equation (PDE), with a mixed boundary condition for the PDE. The coupling appears both in the ODE and in the Neuman boundary condition of the PDE. For this system, initial conditions or state variables and some empirical parameters are assumed to be unknown and need to be estimated. The Lagrangian multiplier method is used to connect the dynamics of the system and the cost function defined as the least square error between the simulation values and the available measurements. The adjoint state method is applied to the objective functional to get the adjoint system and the gradients with respect to parameters and initial state. The cost functional is optimised, employing the steepest descent method to estimate parameters and initial state. This general approach is illustrated by two application examples corresponding to two different landslide models that validate the presented optimal estimation approach. The first one is about state and parameter estimation in an extended sliding-consolidation landslide model, and the second one is in the viscoplastic sliding-consolidation landslide model.