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In this work, we consider the dynamics of repairable systems characterized by three distinct states: one signifying normal operational states, another representing degraded conditions and a third denoting failed conditions. These systems are characterized by their ability to be repaired when failures and/or degradation occur. Typically described by transport equations, these systems exhibit a coupled nature, interlinked through integro-differential equations and integral boundary conditions that dictate the transitions among all the states. In this paper, we address two less-explored facets: 1) the well-posedness and the asymptotic behavior of such systems with maximum repair time being finite; and 2) the bilinear controllability of the system via repair actions. In particular, we focus on the case where only one degraded and one failed states exist. We first discuss part 1) for given time-independent repair rates and then design the space-time dependent repair strategies that can manipulate system dynamics to achieve the desired level over a finite horizon. Our objective is to enhance the system availability -- the probability of being operational when needed over a fixed period of time. We present rigorous analysis and develop control strategies that leverage the bilinear structure of the system model.

Reparable systems are systems that are characterized by their ability to undergo maintenance actions when failures occur. These systems are often described by transport equations, all coupled through an integro-differential equation. In this paper, we address the understudied aspect of the controllability of reparable systems. In particular, we focus on a two-state reparable system and our goal is to design a control strategy that enhances the system availability -- the probability of being operational when needed. We establish bilinear controllability, demonstrating that appropriate control actions can manipulate system dynamics to achieve desired availability levels. We provide theoretical foundations and develop control strategies that leverage the bilinear structure of the equations.

This work develops an efficient and accurate optimization algorithm to study the optimal mixing problem driven by boundary control of unsteady Stokes flows, based on the theoretical foundation laid by Hu and Wu in a series of work. The scalar being mixed is purely advected by the flow and the control is a force exerted tangentially on the domain boundary through the Navier slip conditions. The control design has potential applications in many industrial processes such as rotating wall driven mixing, micromixers with acoustic waves, and artificial cilia mixing. The numerical algorithms have high complexity, high accuracy demand, and high computing expense, due to the multiscale nature of the mixing problem and the optimization requirements. A crucial problem is the computation of the G$\^a$teaux derivative of the cost functional. To this end, a hybrid approach based on variational formula and finite difference is built with high accuracy and efficiency to treat various types of control input functions. We have experimented with various optimization schemes including the steepest descent algorithm, the conjugate gradient method and two line search options (backtracking and exact line search). We are able to identify and implement the best combinations. The numerical simulations show that the mixing efficacy is limited when only one single type of control is applied, but can be enhanced when more diverse control types and more time segmentation are utilized. The mix-norm in the optimal mixings decays exponentially. The numerical study in this work demonstrates that boundary control alone could be an effective strategy for mixing in incompressible flows., Comment: 36 pages

This work develops scientific computing techniques to further the exploration of using boundary control alone to optimize mixing in Stokes flows. The theoretical foundation including mathematical model and the optimality conditions for solving the optimal control has been established by Hu and Wu in a series of work. The scalar being mixed is purely advected by the flow and the control is exerted tangentially on the domain boundary through the Navier slip conditions. The control design is motivated by the physical observations that the moving or rotating walls accelerate mixing. A gradient descent-based optimization algorithm is designed. A critical problem is the computation of the Gateaux derivative or the gradient of the cost functional. Two methods are proposed: one is based on the Variational Formula (VF) and one utilizes Algorithmic Differentiation (AD). The convergence of the algorithm is studied and various designs of boundary control using cosine and sine functions with time segmentation are computed. The algorithm has a first order convergence rate and the VF method is more efficient by taking only one third of the time as the AD method when the dimension of control basis is large. The numerical implementations show that the boundary control produces similar mixing results as internal mixings in the existing literature. The mixing effect becomes better when more diverse basis control functions and more time segmentation are utilized. It is shown that the mixing decay rate in time follows power rules, approximately. The numerical study in this work suggests that boundary control alone could be an effective strategy for mixing in incompressible fluid flows., Comment: This article was significantly upgraded, and the new version is available in arXiv:2306.10690

This paper is concerned with a bilinear control problem for enhancing convection-cooling via an incompressible velocity field. Both optimal open-loop control and closed-loop feedback control designs are addressed. First and second order optimality conditions for characterizing the optimal solution are discussed. In particular, the method of instantaneous control is applied to establish the feedback laws. Moreover, the construction of feedback laws is also investigated by directly utilizing the optimality system with appropriate numerical discretization schemes. Computationally, it is much easier to implement the closed-loop feedback control than the optimal open-loop control, as the latter requires to solve the state equations forward in time, coupled with the adjoint equations backward in time together with a nonlinear optimality condition. Rigorous analysis and numerical experiments are presented to demonstrate our ideas and validate the efficacy of the control designs., Comment: 27 pages, 7 figures, 3 tables

We study temperature and velocity output tracking problem for a two-dimensional room model with the fluid dynamics governed by the linearized translated Boussinesq equations. Additionally, the room model includes finite-dimensional models for actuation and sensing dynamics, thus the complete model dynamics are governed by an ODE-PDE-ODE system. As the main result, we design a low-dimensional internal model based controller for robust output racking of the room model. Efficiency of the controller is demonstrated through a numerical example of velocity and temperature tracking., Comment: 26 pages, 9 figures, submitted

This paper is concerned with the problem of enhancing convection-cooling via active control of the incompressible velocity field, described by a stationary diffusion-convection model. This essentially leads to a bilinear optimal control problem. A rigorous proof of the existence of an optimal control is presented and the first order optimality conditions are derived for solving the control using a variational inequality. Moreover, the second order sufficient conditions are established to characterize the local minimizer. Finally, numerical experiments are conducted utilizing finite elements methods together with nonlinear iterative schemes, to demonstrate and validate the effectiveness of our control design., Comment: 24 pages, 17 figures

We consider robust output regulation of a partial differential equation model describing temperature evolution in a room. More precisely, we examine a two-dimensional room model with the velocity field and temperature evolution governed by the incompressible steady state Navier-Stokes and advection-diffusion equations, respectively, which coupled together form a simplification of the Boussinesq equations. We assume that the control and observation operators of our system are distributed, whereas the disturbance acts on a part of the boundary of the system. We solve the robust output regulation problem using a finite-dimensional low-order controller, which is constructed using model reduction on a finite element approximation of the model. Through numerical simulations, we compare performance of the reduced-order controller to that of the controller without model reduction as well as to performance of a low-gain robust controller., Comment: 12 pages, 5 figures. Accepted for publication in the Proceedings of the 24th International Symposium on Mathematical Theory of Networks and Systems, 23-27 August, 2021

We consider an approximating control design for optimal mixing of a non-dissipative scalar field $\theta$ in unsteady Stokes flows. The objective of our approach is to achieve optimal mixing at a given final time $T>0$, via the active control of the flow velocity $v$ through boundary inputs. Due to the zero diffusivity of the scalar field $\theta$, establishing the well-posedness of its G\^{a}teaux derivative requires $\sup_{t\in[0,T]}\|\nabla \theta\|_{L^2}<\infty$, which in turn demands the flow velocity field to satisfy the condition $ \int^{T}_{0}\|\nabla v\|_{L^{\infty}(\Omega)}\, dt<\infty$. This condition results in the need to penalize the time derivative of the boundary control in the cost functional. As a result, the optimality system becomes difficult to solve \cite{hu2017boundary}. Our current approximating approach will provide a more transparent optimality system, with the set of admissible controls being $L^{2}$ in both time and space. This is achieved by first introducing a small diffusivity to the scalar equation and then establishing a rigorous analysis of convergence of the approximating control problem to the original one as the diffusivity approaches to zero. Uniqueness of the optimal solution is obtained for the two dimensional case.

We discuss the problem of optimal impulse control representing the preventive maintenance of a simple reparable system. The system model is governed by coupled transport and integro-differential equations in a nonreflexive Banach space. The objective of this paper is to construct nonnegative impulse control inputs at given system running times that minimize the probability of the system in failure mode. To guarantee the nonnegativity of the controlled system, we consider the control inputs to depend on the system state. This essentially leads to a bilinear control problem. We first present a rigorous proof of existence of an optimal controller and then apply the variational inequality to derive the first-order necessary conditions of optimality.