An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise
We study a linear quadratic problem for a system governed by the heat equation on a halfline with Dirichlet boundary control and Dirichlet boundary noise. We show that this problem can be reformulated as a stochastic evolution equation in a certain weighted space. An appropriate choice of weight allows us to prove a stronger regularity for the boundary terms appearing in the infinite dimensional state equation. The direct solution of the Riccati equation related to the associated non-stochastic problem is used to find the solution of the problem in feedback form and to write the value function of the problem.
Key words: heat equation, Dirichlet boundary conditions, boundary noise, boundary control, weighted space, analytic semigroup, stochastic convolution, linear quadratic control problem, Riccati equation.
MSC 2000: 35G15, 37L55, 49N10.
In this paper we are concerned with a linear quadratic control problem for a heat equation on the halfline with Dirichlet boundary control and boundary noise. More precisely, for fixed , we deal with the equation
where is a one dimensional Brownian motion and is a square-integrable control. Let us recall that a deterministic boundary control problem
and it is easy to show that for all . Therefore, the process
seems to be a good candidate for a solution to (1). However, it was shown in [DaPratoZabczyk93] that the process is not -valued. More precisely, it was shown that the solution to (1) considered on a finite interval and for , when rewritten in the form (3), is well defined in a negative Sobolev space for only. It is easy to see that the same conclusion holds in the case of halfline. Then it was shown in [AlosBonaccorsi02], see also [BonaccorsiGuatteri02], that the process can be defined pointwise on and it takes values in a weighted space . This fact was used to study some properties of the process (in fact in the aforementioned papers more general nonlinear equations are studied) but the problem is not reformulated as a stochastic evolution equation in and therefore advantages of using the weighted space are somewhat limited.
Following the idea of Krylov [Krylov01] we introduce the weighted spaces , where for we have
It was proved in [Krylov99] and [Krylov01] that the Dirichlet Laplacian defined on extends to a generator of an analytic semigroup on . We will show that the Dirichlet map takes values in for a certain and therefore equation (3), when considered in , can be given a form
that is, we will study a controlled evolution equation
for . This fact is a starting point for our analysis of the linear quadratic control problem (1). We will demonstrate that the control problem (4) when considered in the space can be solved using classical by now techniques presented, for example, in [BDDM2]. Let us emphasize that while focus of this paper is on the most interesting case of boundary control and boundary noise a more general control problem
with spatially distributed noise and control might be easily considered using the same technique.
Let us note that if the boundary conditions are of Neumann type then the analogue of equation (1) has a solution in and has been studied intensely (also for more general parabolic equations with boundary noise), see for example [DaPratoZabczyk93], [Maslowski95], [DebusscheFuhrmanTessitore07], [dunmas].
We study the linear quadratic problem characterized by the cost functional
and governed by a state equation of the form (4). the operator that appears in (6) is in for a certain Hilbert space and is symmetric and positive. The direct solution of the Riccati equation related to a linear quadratic problem driven by a stochastic equation different from ours was studied in the Neumann case (non-weighted setting) in [Flandoli86] (see also [Ahmed81] and [DaPrato84] for the control inside the domain case ()). Our approach is different from the one used in the aforementioned works since we directly use the solution of the Riccati equation for the “associated” deterministic problem.
The deterministic linear quadratic problem associated to ours is that characterized by the state equation
and the functional
It is well known, see [BDDM2] and Section 3 below, that the solution to the linear quadratic problem given above is determined by the operator-valued function which solves the so-called Riccati equation
Such a problem has been intensely studied (see [BDDM2] and [LasieckaTriggiani00] and the references therein). We will refer in particular to the direct solution approach and we will use the formalism introduced in Section 2.2. of [BDDM2]. We show that the Riccati equation (8) has a unique solution in the space (see Definition 3.2). Let us note that in the deterministic case the minimum of the cost functional (7) is given by .
In the study of the problem with boundary noise some of the tools and the results of the deterministic case, as the properties of the elements of and the solution of (8), are still useful. It is possible to express the value function and the optimal feedback in terms of . A term due to the noise appears in the expression of the minimal cost and we have that (Theorem 3.7):
2 The heat equation in
We will work in a weighted space , where either or for some and . All the results proved in the sequel are valid for both weights and therefore, in order to simplify notations we will use the same notation for both weights. Let us recall that if and only if
and is a Hilbert space with the scalar product
Given , the Dirichlet map is defined as follows:
It is well known that for every the solution to the heat equation with zero Dirichlet boundary condition
is given by the following well known expression
This formula defines the corresponding heat semigroup in . It is also well known that is a symmetric -semigroup of contractions on .
2.2 Properties of the heat semigroup on
For each of the weights considered above, the heat semigroup extends to a bounded semigroup on with generator . The semigroup is analytic.
The case : .
Let . Then by Theorem 2.5 in [Krylov01] there exists independent of and such that
Since is dense in , can be extended to and the strong continuity follows by standard arguments. Let be the generator of in and let . Clearly
and is dense in . Therefore is a core for the generator of in . If then
and again by Theorem 2.5 in [Krylov01] we have
Since is a core for the generator in , the above estimate can be extended to any and therefore
The last inequality is equivalent to the analyticity of the semigroup in .
The case : .
Let and . Then the functions and are in and for both weights . It follows that
for a certain . The fact that does not depend on is a consequence of the property of on (showed in the first part of the proof) and on . Therefore has an extension to a semigroup on and the -property follows by standard arguments. Similar arguments yield analyticity of . ∎
Assume that and . Then
In particular for all .
We consider the case of only. The other case may be proved by similar if somewhat simpler arguments.
Note first that if then for all 333 denotes the real interpolation space, see for example Theorem 11.5.1 in [MartinezAlix01]. Hence the claim will follow if we show that for
By Theorem 10.1 of [LionsMagenes72]) if and only if
and taking into account (15) it is enough to show that
where , and are respectively
Since for we have we find that converges for every . can be estimated, using that the standard estimate
where the finiteness of the first term follows from (15). The estimate for can be obtained in a similar way. ∎
2.3 Properties of the solution of the state equation
Let be a real Brownian motion on a probability space and let denote the natural filtration of . We need to give a rigorous meaning to equation (1). To this end we will assume in the sequel that
are fixed. We will denote by the operator and . By Proposition 2.1 the semigroup is analytic and therefore for any
see for example [Pazy83] (Theorem 6.13 page 75). By Lemma 2.2 the operator is bounded444For the space is defined as a completion of with respect to the norm . Moreover, for the operator
is bounded as well. We will write Now, we reformulate equation equation (1), still formally, as a stochastic evolution equation in :
where the control is chosen in the set of progressively measurable processes endowed with the norm
The next two results show that we can give a meaning to (19).
For all the following holds.
(i) The operator is bounded for each and the function
is continuous for every .
(iii) For every the process
is well defined, belongs to and has continuous trajectories in .
(i) It follows immediately from the definition of and Lemma 2.2 since
Such an estimate gives also, through standard arguments, the mean square continuity. The continuity follows from (20) for using a factorization argument as in [DaPratoZabczyk93] Theorem 2.3 page 174. ∎
Let be fixed, and . Then the process
is well defined, , and there exists such that
Moreover, is in and has continuous trajectories.
Let . An -valued predictable process , defined on is called a mild solution of (19) if
2.4 The approximating equation
Let . We will approximate using
We have that
We use it to obtain more regularity and to guarantee the existence of a strong solution and then to be able to apply the Ito’s rule (Proposition 3.6). From Proposition 2.6 we know that . We have and then . Furthermore, satisfies the following stochastic differential equation:
in strong (an then mild) sense (see [DaPratoZabczyk92] Section 6.1). So we have
3 The linear quadratic problem
Let us recall that we work under the assumption
We consider another Hilbert space , an operator and a symmetric and positive . For a fixed we define the set of the admissible controls as . We consider the linear quadratic optimal control problem governed by equation (19) and quadratic cost functional (to be minimized)
The value function of the problem is
We consider now the “associated” deterministic linear quadratic problem. It is characterized by the state equation
by the set of admissible controls and by the functional
In what follows we we will use the following notations.
Note that ([BDDM2] page 137) for every
The Riccati equation formally associated with the deterministic control problem (28) has the form
but the concept of solution to this equation requires a rigorous definition. We start with some notations.
We denote by the set of all such that
Given , the norm is defined as
It can be proved (see [BDDM2] page 205) that , endowed with the norm , is a Banach space. We will use the notation .
Note that if then (since )
We say that is a weak solution of the Riccati equation (30) if for all and all
We recall now the existence and uniqueness theorem for the (32):
See [BDDM2] Theorem 2.1 page 207 for the proof of (i) and [BDDM2] Proposition 2.1 page 206 for (ii). ∎
3.1 Dynamic Programming
We have that
We use the fact that satisfies the mild equation (33). We have that