Optimization of nonlocal time-delayed feedback controllers ††thanks: This work was supported by DFG in the framework of the Collaborative Research Center SFB 910, projects A1 and B6.
A class of Pyragas type nonlocal feedback controllers with time-delay is investigated for the Schlögl model. The main goal is to find an optimal kernel in the controller such that the associated solution of the controlled equation is as close as possible to a desired spatio-temporal pattern. An optimal kernel is the solution to a nonlinear optimal control problem with the kernel taken as control function. The well-posedness of the optimal control problem and necessary optimality conditions are discussed for different types of kernels. Special emphasis is laid on time-periodic functions as desired patterns. Here, the cross correlation between the state and the desired pattern is invoked to set up an associated objective functional that is to be minimized. Numerical examples are presented for the 1D Schlögl model and a class of simple step functions for the kernel.
Keywords:Schlögl model, Nagumo equation, Pyragas type feedback control, nonlocal delay, controller optimization, numerical method
In this paper, we consider a class of nonlocal feedback controllers with application to the control of certain nonlinear partial differential equations. The research on feedback control laws of this type has become quite active in theoretical physics for stabilizing wave-type solutions of reaction-diffusion systems such as the Schlögl model (also known as Nagumo or Chafee-Infante equation) or the FitzHugh-Nagumo system.
The controllers can be characterized as follows: First of all, they are a generalization of Pyragas type controllers that became very popular in the past. We refer to pyragas1992 (), pyragas2006 (), and the survey volume schoell_schuster2008 (). In the simplest form of Pyragas type feedback control, the difference of the current state and the retarded state , multiplied with a real number , is taken as control, i.e. the feedback control is
where is a fixed time delay and is the feedback gain.
In the nonlocal generalization we consider in this paper, the feedback control is set up by an integral operator of the form
Here, different time delays appear in a distributed way. Depending on the particular choice of the kernel , various spatio-temporal patterns of the controlled solution can be achieved. We refer to bachmair_schoell14 (); loeber_etal2014 (); siebert_schoell14 (), and siebert_alonso_baer_schoell14 () with application to the Schlögl model and to atay2003 (); kyrichenko_blyuss_schoell2014 (); wille_lehnert_schoell2014 () with respect to control of ordinary differential equations.
Our main goal is the selection of the kernel in an optimal way. We want to achieve a desired spatio-temporal pattern for the resulting state function and look for an optimal feedback kernel to approximate this pattern as closely as possible. For this purpose, in the second half of the paper we will concentrate on a particular choice of as a step function.
We are optimizing feedback controllers but we shall apply methods of optimal control to achieve our goal. This leads to new optimal control problems for reaction-diffusion equations containing nonlocal terms with time delay in the state equation. We develop the associated necessary optimality conditions and discuss numerical approaches for solving the problems posed. Working on this class of problems, we observed that standard quadratic tracking type objective functionals are possibly not the right tool for approximating desired time-periodic patterns. We found out that the so-called cross correlation partially better fits to our goals. We report on our numerical tests at the end of this paper.
This research contributes results to the optimal control of nonlinear reaction diffusion equations, where wave type solutions such as traveling wave fronts or spiral waves occur in unbounded domains. We mention the papers borzi_griesse06 (); brandao_etal08 (); ckp09 () on the optimal control of systems that develop spiral waves or kun_nag_cha_wag2011 (); kunisch_wagner2012-3 (); kunisch_wang12 () on systems with heart medicine as background. Moreover, we refer to buch_eng_kamm_tro2013 (); casas_ryll_troeltzsch2014b (), where different numerical and theoretical aspects of optimal control of the Schlögl or FitzHugh-Nagumo equations are discussed. It is a characteristic feature of such systems that the computed optimal solutions might be unstable with respect to perturbations in the data, in particular initial data.
Feedback control aims at generating stable solutions. Various techniques of feedback control are known, we refer only to the monographies coron07 (); lasiecka_triggiani2000a (); lasiecka_triggiani2000b (); Krstic2010 () and to the references cited therein. Moreover, we mention gugat_troeltzsch2013 () on feedback stabilization for the Schlögl model. Pyragas type feedback control is one associated field of research that became very active, cf. schoell_schuster2008 () for an account on current research in this field. In associated publications, the feedback control laws were considered as given. For instance, the kernel in nonlocal delayed feedback was given and it was studied what kind of patterns arise from different choices of the kernel.
The novelty of our paper is that we study an associated inverse (say design) problem: Find a kernel such that the associated feedback solution best approximates a desired pattern.
2 Two models of feedback control
We consider the following semilinear parabolic equation with reaction term and control function (forcing) ,
subject to appropriate initial and boundary conditions in a spatio-temporal domain . Using a feedback control in the form (1), we arrive at the following nonlinear initial-boundary value problem that includes a nonlocal term with time delay,
for almost all .
Here, denotes the outward normal derivative on . We want to determine a feedback kernel such that the solution to (2) is as close as possible to a desired function . The function will have to obey certain restrictions, namely
where is a given (large) positive constant. This upper bound is chosen to have a uniform bound for . It is needed for proving the solvability of the optimal control problem.
We shall present the main part of our theory for the general type of defined above. In our numerical computations, however, we will concentrate on functions of the following particular form: We select such that , and define
In (6), we will also vary in the state equation as part of the control variables to be optimized. In contrast to this, is assumed to be fixed in the model with a general control function . In the special model, we have a restricted flexibility in the optimization, because only the real numbers can be varied. Yet, we are able to generate a class of interesting time-periodic patterns.
Throughout the paper we will rely on the following
Assumptions. The set , , is a bounded Lipschitz domain; for , we set . By , a finite terminal time is fixed. In theoretical physics, also the choice is of interest. However, we do not investigate the associated analysis, because an infinite time interval requires the use of more complicated function spaces. Moreover, the restriction to a bounded interval fits better to the numerical computations. Throughout the paper, we use the notation and . for the space-time cylinder.
We will often use the term ”wave type solution” or ”traveling wave”. This is a function that can be represented in the form with some other smooth function . Here, is the velocity of the wave type solution. Such solutions are known to exist in but not in in a bounded interval .
In our paper, the terms ” wave type solution” or ”traveling wave” stand for solutions of the Schlögl model in the bounded domain . We use these terms, since the computed solutions exhibit a similar behavior as associated solutions in .
The reaction term is defined by
where and are fixed real numbers. In our computational examples, we will take . The numbers , , define the fixed points of the (uncontrolled) Schlögl model (1). In view of the time delay, we have to provide initial values for in the interval for the general model (2) and in for the special model (6). We assume or , respectively. The desired state is assumed to be bounded and measurable on .
3 Well-posedness of the feedback equation
In this section, we prove the existence and uniqueness of a solution to the general feedback equation (2). To this aim, we first reduce the equation to an inhomogeneous initial-boundary value problem. For , we write
The function is associated with the fixed initial function and is defined by
notice that we have in the integral above. By the assumed continuity of , the function belongs to .
Next, for given , we introduce a linear integral operator by
Substituting , we obtain the equivalent representation
Inserting and in the state equation (2), we obtain the following nonlocal initial-boundary value problem:
In the next theorem, we use the Sobolev space
For all , and , the problem (9) has a unique solution .
We use the same technique that was applied in casas_ryll_troeltzsch2014 () to show the existence and continuity of the solution to the FitzHugh-Nagumo system. Let us mention the main steps. First, we apply a simple transformation that is well-known in the theory of evolution equations. We set
with some . This transforms the partial differential equation in (9) to an equation for the new unknown function ,
where the integral operator is defined by
If , then both operators and are continuous linear operators in , for all . Moreover, due to the factor , the norm of tends to zero as . We obtain
with some constant . To have this estimate, we assumed in (3) that is uniformly bounded by the constant . If is sufficiently large, then we have
because the coercive term in the left side is dominating the other terms, cf. casas_ryll_troeltzsch2014 ().
With this inequality, an a priori estimate can be derived in for any solution of the equation (9). Now, we can proceed as in casas_ryll_troeltzsch2014 (): A fixed-point principle is applied in to prove the existence and uniqueness of the solution that in turn implies the same for . For the details, the reader is referred to casas_ryll_troeltzsch2014 (), proof of Theorem 2.1. However, we mention one important idea: Thanks to (11), the term absorbes the non-monotone terms in the equation (10) so that, in estimations, equation (10) behaves like the parabolic equation
with a monotone non-decreasing nonlinearity and given right-hand side , . This fact can be exploited to verify, for each , the existence of a constant with the following property: If obeys and is the associated solution to (2), then
4 Analysis of optimization problems for feedback controllers
4.1 Definition of two optimization problems
General kernel as control
Let a desired function be given. In our later applications, models a desired spatio-temporal pattern. Moreover, we fix a non-negative function . This function is used for selecting a desired observation domain. We consider the feedback equation (2) and want to find a kernel such that the associated solution approximates as close as possible in the domain of observation. This goal is expressed by the following functional that is to be minimized,
Here, is a Tikhonov regularization parameter. The standard choice of is for all . Another selection will be applied for periodic functions : for all with and for all with .
By Theorem 3.1, to each there exists a unique associated state function that will be denoted by . Then does only depend on and we obtain the reduced objective functional ,
Therefore, our general optimization problem can be formulated as follows:
where is the convex and closed set defined by
Notice that is a weakly compact subset of . The restrictions on are motivated by the background in mathematical physics. In particular, the restriction on the integral of guarantees that
if in Q. By the definition of , the optimization is subject to the state equation (2).
Special kernel as control
The other optimization problem we are interested in, uses the particular form (5) of the kernel ,
where is the solution of (6) for a given triplet . This problem might fail to have an optimal solution, because the set of admissible triplets is not closed. Notice that we need in (6). Therefore, we fix and define the slightly changed admissible set
that is compact. In this way, we obtain the special finite-dimensional optimization problem for step functions ,
4.2 Discussion of (PG)
The control-to-state mapping
Next, we discuss the differentiability of the control-to-state mappings and . First, we consider the case of the general kernel . The analysis for the particular kernel (5) is fairly analogous but cannot deduced as a particular case of (PG). We will briefly sketch it in a separate section.
By Theorem 3.1, we know that the mapping is well defined from to . Now we discuss the differentiability of . To slightly simplify the notation, we introduce an operator by
where was introduced in (8); notice that is bilinear. Let us first show the differentiability for .
We fix , and select varying increments , . Then we have
where is a linear continuous operator and is a remainder term. It is easy to confirm that
Therefore, is Fréchet-differentiable. As a continuous bilinear form, is also of class .
Now, we investigate the control-to-state mapping defined by where the state function is defined as the unique solution to
In what follows, the initial function will be kept fixed and is therefore not mentioned. Of course, and some of the operators below depend on , but we will not explicitely mention this dependence. To discuss , we need known properties of the following auxiliary mapping , where
Now (consider as given and keep the initial function fixed), solves (12) if and only if , i.e.
We introduce a new mapping defined by
Then, (13) is equivalent to the equation
We have proved above that the mapping is of class from to . Obviously, also the linear mapping is of class from to . By the chain rule, also is from and the mappings , are continuous in the associated pairs of spaces.
To use the implicit function theorem, we prove that is continuously invertible at any fixed pair . Therefore, we consider the equation
with given right-hand side and show the existence of a unique solution . The equation is equivalent with
Writing for convenience , we obtain the simpler form
A function does not in general belong to . To overcome this difficulty, we set and transform the equation to
where . As the next result shows, is the solution of a parabolic PDE, hence .
Let with be given. Then we have if and only if solves
where is the solution associated with , i.e.
We refer to casas_ryll_troeltzsch2014 (). For monotone non-decreasing functions , this result is well known in the theory of semilinear parabolic control problems, see e.g. cas93 (), rayzid98 (), or (tro10book, , Thm. 5.9). By Lemma 4.1, the solution of (17) is the unique solution of the linear PDE
subject to and homogeneous Neumann boundary conditions. By the same methods as above we find that, for all , equation has a unique solution .
After transforming back by , we have found that for all , (15) has a unique solution given by . Therefore, the inverse operator exists. The continuity of this inverse mapping follows from a result of casas_ryll_troeltzsch2014 () that the mapping defined by (18) is continuous in .
Next, we consider the operator . It exists by the chain rule and admits the form
Setting again and , we see that
where, by Lemma 4.1, solves the equation
subject to homogeneous initial and boundary conditions. Therefore, is the unique solution to
By for , we can re-write this as
Again, the mapping is continuous from to .
Collecting the last results, we have the following theorem:
Theorem 4.1 (Differentiability of )
The control-to-state mapping associated with equation (9) is of class . The first order derivative is obtained as the unique solution to
We already know by Theorem 3.1 that, for all , there exists a unique solution solving the equation
We discussed above that the assumptions of the implicit function theorem are satisfied. Now this theorem yields that the mapping is of class .
The derivative is obtained by implicit differentiation. By definition of , we have
Implicit differentiation yields that is the unique solution of (19). Notice that
4.3 Existence of an optimal kernel
For all , (PG) has at least one optimal solution .
Let with for all be a minimizing sequence. Since is bounded, convex, and closed in , we can assume without limitation of generality that converges weakly in to , i.e. , . The associated sequence of states obeys the equations
By the principle of superposition, we split the functions as , where is the solution of (21) with right-hand side and initial value , while is the solution to the right-hand side defined above and zero initial value. In view of (12), all state functions , hence also the functions , are uniformly bounded in . Thanks to (DiBenedetto1986, , Thm. 4), the sequence is bounded in some Hölder space . By the Arzela-Ascoli theorem, we can assume (selecting a subsequence, if necessary) that converges strongly in . Adding to the fixed function , we have that converges strongly to some in .
The boundedness of also induces the boundedness of the sequence in , in particular in . Therefore, we can assume that converges weakly in to , . Since is the sequence of solutions to the ”linear” equation (21) with right-hand side , the weak convergence of induces the weak convergence of in , where solves (21) with right-hand side .
Finally, we show that
so that is the state associated with . Obviously, it suffices to prove that converges weakly to in . To this aim, let an arbitrary be given. Then we have
Clearly, the strong convergence of in yields
in . Along with the weak convergence of , this implies
In view of (22), we finally arrive at
as . Since this holds for arbitrary , this is equivalent to the desired weak convergence in .
4.4 Necessary optimality conditions
4.4.1 Adjoint equation
In the next step of our analysis, we establish the necessary optimality conditions for a (local) solution of the optimization problem (PG). This optimization problem is defined by
Although the admissible set belongs to , we consider this as an optimization problem in the Hilbert space .
To set up associated necessary optimality conditions for an optimal solution of (23), we first determine a useful expression for the derivative of the objective functional . We have
Let now be an arbitrary (i.e. not necessarily optimal) be given and let be the associated state. Then we obtain for