We show how a posteriori goal oriented error estimation can be used to efficiently solve the subproblems occurring in a Model Predictive Control (MPC) algorithm. In MPC, only an initial part of a computed solution is implemented as a feedback, which motivates grid refinement particularly tailored to this context. To this end, we present a truncated cost functional as objective for goal oriented adaptivity and prove under stabilizability assumptions that error indicators decay exponentially outside the support of this quantity. This leads to very efficient time and space discretizations for MPC, which we will illustrate by means of various numerical examples.
This paper considers optimal control of fractional parabolic PDEs with both state and control constraints. The key challenge is how to handle the state constraints. Similarly, to the elliptic case, in this paper, we establish several new mathematical tools in the parabolic setting that are of wider interest. For example, existence of solution to the fractional parabolic equation with measure data on the right-hand-side. We employ the Moreau-Yosida regularization to handle the state constraints. We establish convergence, with rate, of the regularized optimal control problem to the original one.…
We propose an approximation scheme for a class of semilinear parabolic equations that are convex and coercive in their gradients. Such equations arise often in pricing and portfolio management in incomplete markets and, more broadly, are directly connected to the representation of solutions to backward stochastic differential equations. The proposed scheme is based on splitting the equation in two parts, the first corresponding to a linear parabolic equation and the second to a Hamilton-Jacobi equation. The solutions of these two equations are approximated using, respectively, the Feynman-Kac …
The tangential condition was introduced in [Hanke et al., 95] as a sufficient condition for convergence of the Landweber iteration for solving ill-posed problems. In this paper we present a series of time dependent benchmark inverse problems for which we can verify this condition.
This paper presents a maximum principle-based approach in the establishment of input-to-state stability (ISS) for a class of nonlinear parabolic partial differential equations (PDEs) over higher dimensional domains with variable coefficients and different types of nonlinear boundary conditions. Technical development on ISS analysis of the considered systems is detailed, and an example of establishing ISS estimates for a nonlinear parabolic equation with, respectively, a nonlinear Robin boundary condition and a nonlinear Dirichlet boundary condition is provided to illustrate the application of …
In this paper, we address stability of parabolic linear Partial Differential Equations (PDEs). We consider PDEs with two spatial variables and spatially dependent polynomial coefficients. We parameterize a class of Lyapunov functionals and their time derivatives by polynomials and express stability as optimization over polynomials. We use Sum-of-Squares and Positivstellensatz results to numerically search for a solution to the optimization over polynomials. We also show that our algorithm can be used to estimate the rate of decay of the solution to PDE in the L2 norm. Finally, we validate the …
This paper considers the backstepping state feedback control of coupled linear parabolic PDEs with spatially varying coefficients and bilateral actuation. By making use of the folding technique, a system representation with unilateral actuation is obtained, allowing to apply the standard backstepping transformation. To ensure the regularity of the solution, the folded system is subject to unusual folding boundary conditions, which lead to additional boundary couplings between the PDEs. Therefore, the solution of the corresponding kernel equations determining the transformations is a very chall…
In this paper, we consider a parabolic PDE on a torus of arbitrary dimension. The nonlinear term is a smooth function of polynomial growth of any degree. In this general setting, the corresponding Cauchy problem is not necessarily well posed. We show that the equation in question is approximately controllable by only a finite number of Fourier modes. This result is proved by using some ideas from the geometric control theory introduced by Agrachev and Sarychev.
This paper provides novel Input-to-State Stability (ISS)-style maximum principle estimates for classical solutions of highly nonlinear 1-D parabolic Partial Differential Equations (PDEs). The derivation of the ISS-style maximum principle estimates is performed by using an ISS Lyapunov Functional for the sup norm. The estimates provide fading memory ISS estimates in the sup norm of the state with respect to distributed and boundary inputs. The obtained results can handle parabolic PDEs with nonlinear and non-local in-domain terms/boundary conditions. Three illustrative examples show the efficie…
In this paper, we introduce the notion of relative $\mathcal{K}$-equi-stability (RKES) to characterize the uniformly continuous dependence of (weak) solutions on external disturbances for nonlinear parabolic PDE systems. Based on the RKES, we prove the input-to-state stability (ISS) in the spatial sup-norm for a class of nonlinear parabolic PDEs with either Dirichlet or Robin boundary disturbances. Two examples, concerned respectively with a super-linear parabolic PDE with Robin boundary condition and a $1$-D parabolic PDE with a destabilizing term, are provided to illustrate the obtained ISS …
Optimal control of the singular nonlinear parabolic PDE which is a distributional formulation of multidimensional and multiphase Stefan-type free boundary problem is analyzed. Approximating sequence of finite-dimensional optimal control problems is introduced via finite differences. Existence of the optimal control and the convergence of the sequence of discrete optimal control problems both with respect to functional and control is proved. In particular, convergence of the method of finite differences, and existence, uniqueness and stability estimations are established for the singular PDE pr…
We study and compare two concepts for weak solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). The first is that of mild solutions as it appears, e.g., in the log-Laplace functionals of historical superprocesses. The aim of this paper is to show that mild solutions are also solutions in a viscosity sense. This result is motivated by the fact that mild solutions can provide value functions and optimal strategies for problems of stochastic optimal control. Since unique mild solutions exist under weak conditions, we obtain as a corollary a general existence re…
We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary disturbances are monotone control systems. With these two facts, we establish that ISS of the original nonlinear parabolic PDE with constant boundary disturbances is equivalent to ISS of a closely related nonlinear parabolic PDE with constant distributed d…
Finite element approximations of Dirichlet boundary control problems governed by parabolic PDEs on convex polygonal domains are studied in this paper. The existence of a unique solution to optimal control problems is guaranteed based on very weak solution of the state equation and $L^2(0,T;L^2(Γ))$ as control space. For the numerical discretization of the state equation we use standard piecewise linear and continuous finite elements for the space discretization of the state, while a dG(0) scheme is used for time discretization. The Dirichlet boundary control is realized through a space-time $L…
We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary disturbances are monotone control systems. With these two facts, we establish that ISS of the original nonlinear parabolic PDE with constant \textit{boundary disturbances} is equivalent to ISS of a closely related nonlinear parabolic PDE with constant \tex…
For an arbitrary parameter $p\in [1,+\infty]$, we consider the problem of exponential stabilization in the spatial $L^{p}$-norm, and $W^{1,p}$-norm, respectively, for a class of anti-stable linear parabolic PDEs with space-time-varying coefficients in the absence of a Gevrey-like condition, which is often imposed on time-varying coefficients of PDEs and used to guarantee the existence of smooth (w.r.t. the time variable) kernel functions in the literature. Then, based on the obtained exponential stabilities, we show that the solution of the considered system depends continuously on the $L^{p}$…
We analyze the sensitivity of the extremal equations that arise from the first order necessary optimality conditions of nonlinear optimal control problems with respect to perturbations of the dynamics and of the initial data. To this end, we present an abstract implicit function approach with scaled spaces. We will apply this abstract approach to problems governed by semilinear PDEs. In that context, we prove an exponential turnpike result and show that perturbations of the extremal equation's dynamics, e.g., discretization errors decay exponentially in time. The latter can be used for very ef…
To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equation and a backward-in-time adjoint equation to evaluate the reduced gradient in each iteration of the optimization method. In this study, we investigate the use of the parallel-in-time method PFASST in the setting of PDE constrained optimization. In order to develop an efficient fully time-parallel algorithm we discuss different options for applying PFASST to adjoint gradient com…
A De Giorgi Iteration-based Approach for the Establishment of ISS Properties of a Class of Semi-linear Parabolic PDEs with Boundary and In-domain Disturbances
We study constant input delay compensation by using finite-dimensional observer-based controllers in the case of the 1D heat equation. We consider Neumann actuation with nonlocal measurement and employ modal decomposition with $N+1$ modes in the observer. We introduce a chain of $M$ sub-predictors that leads to a closed-loop ODE system coupled with infinite-dimensional tail. Given an input delay $r$, we present LMI stability conditions for finding $M$ and $N$ and the resulting exponential decay rate and prove that the LMIs are always feasible for any $r$. We also consider a classical observer-…
The present paper addresses the topic of boundary output feedback stabilization of parabolic-type equations, governed by linear differential operators which can be diagonalized by the introduction of adequate weighting functions (by means of the Sturm-Liouville method), and which evolve in bounded spatial domains that are subsets of $\mathbb{R}^d,\ d=1,2,3$. Combining ideas inspired by \cite{lhachemi2022finite} for the boundary output feedback control of 1-D parabolic PDEs and \cite{munteanu2019boundary} for the state feedback control of multi-D parabolic PDEs, we report in this paper an outpu…
We prove the well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter $t$ on top of the temporal and spatial variables $(s,y)$ and thus the problem could be considered as a flow of equations. The nonlocality comes from the dependence on the unknown function and its first- and second-order derivatives evaluated at not only the local point $(t,s,y)$ but also at the diagonal line of the time domain $(s,s,y)$. Such equations arise from time-in…
