Mild solutions of semilinear elliptic equations
in Hilbert spaces
This paper extends the theory of regular solutions ( in a suitable sense) for
a class of semilinear elliptic equations in Hilbert spaces.
The notion of regularity is based on the concept of -derivative, which is introduced and discussed. A result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property, that is, it maps continuous functions into -differentiable ones. The validity of this smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature.
It is shown that the results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to
infinite horizon optimal stochastic control problems in infinite dimension
and that, in particular, they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now.
Key words: Elliptic equations in infinite dimension, transition semigroups, optimal control of stochastic PDEs, HJB equations.
AMS classification: 35R15, 65H15, 70H20.
Acknowledgements. The authors are very grateful to Mauro Rosestolato and Andrzej Swiech for valuable discussions and suggestions. In particular, they are indebted to Mauro Rosestolato for Remark 4.3.
- 1 Introduction
- 2 Preliminaries
- 3 Mild solutions of semilinear elliptic equations in Hilbert spaces
- 4 -smoothing properties of transition semigroups
- 5 Application to stochastic control problems
Semilinear elliptic equations with infinitely many variables are an important subject due to their application to time homogeneous stochastic optimal control problems and stochastic games problems over an infinite horizon. The infinite dimensionality of the variables arises in many applied problems, e.g., when the dynamics of the state variables is driven by a stochastic delay equation or by a stochastic PDE. In these cases the resulting Hamilton-Jacobi-Bellman (HJB) equation (associated to the control problem) or Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation (associated to the game) are elliptic equations in infinite dimension.
Only few papers were devoted to study such kind of elliptic equations in the literature, using mainly three approaches, as follows (see the forthcoming book  for a survey of the present literature).
The viscosity solution approach, introduced first, for the second order infinite dimensional case, in [55, 56, 57] and then developed in [70, 71, 51] and, later, for more specific problems in [48, 46, 47, 52], among others. On one hand, this approach allows to cover a big variety of elliptic equations in Hilbert spaces, including fully nonlinear ones; on the other hand, a regularity theory for viscosity solutions is not available in infinite dimension. Viscosity solutions have been employed to treat elliptic elliptic equations only in few papers; in particular, we mention .
The mild solution approach by means of representation of solutions through backward stochastic differential equations (BSDEs). In infinite dimension it was introduced in  (for the parabolic case) and in  (for the elliptic case). This method is applicable, so far, only to semilinear equations satisfying a structural condition on the operators involved and allows to find solutions with a -type regularity when the data are accordingly regular. Moreover, it is suitable to solve the associated control problems in the HJB case. The required structural condition, in the HJB case, substantially states that the control can act on the system modifying its dynamics at most along the same directions along which the noise acts. This may be a stringent requirement preventing the use of this method to solve some important applied problems, e.g. boundary control problems (with the exception of the boundary noise case, see [23, 64]).
The mild solution approach by means of fixed point arguments — the method used here. This method has been introduced first in [15, 50] and then developed in [6, 7] and in various other papers (see e.g. [39, 40, 45, 11, 13, 38, 41, 42, 60, 61, 62, 63]333Similar results, but using a different method based on a convex regularization procedure, were obtained in earlier papers [1, 2, 3] in the special case of convex data and quadratic Hamiltonian function .. Such method, suitable for semilinear equations, consists in proving first smoothing properties of the transition semigroup associated to the linear part of the equation and then applying fixed point theorems. In this way, one finds solutions with -type regularity properties, which allow, in some cases, to solve the associated control problems. Within this approach, the elliptic case has been treated in the papers [7, 45, 13, 38, 62].
The main purpose of this paper is the develop a general framework for the application of the mild solution approach in the elliptic case and to show that such framework allows:
on the other side, to cover HJB equations arising in control problems, like the boundary control ones, which so far cannot be solved by means of other techniques.
We now present the equation we deal with and explain briefly the main ideas. We consider the following class of semilinear elliptic equations in a real separable Hilbert space :
Here , the operator is a linear (possibly unbounded) operator on , and the functions , (where denotes the set of bounded nonnegative linear operators on ), and are measurable. Such equations includes HJB equations associated to discounted time homogeneous stochastic optimal control problems in over infinite time horizon (see Section 5); in this case is called Hamiltonian. Here, our main focus is on the application to this latter case. However, the main results are proved in a more general framework that allows to cover also other cases like HJBI equations associated to differential games.
The type of solutions we study here are called mild solutions, in the sense that they solve the equation in the following integral form:
where is the transition semigroup associated to the linear part of (1.1), that is, to the operator
Note that, in (1.2) only the gradient of appears. However, as it usually arises in applications to control problems, the dependence of the nonlinear part on the gradient can occur in a special form, that is, through a family of linear, possibly unbounded, operators ; this leads to consider a generalized concept of gradient, which we call -gradient and denote by (see Subsection 2.3 for details). Hence, we actually have a nonlinear term in the form , where is a suitable function. We prove existence and uniqueness of solutions to (1.2), with replaced by , by applying a fixed point argument and assuming, to let the method work, a suitable smoothing property of the transition semigroup . The good news are that, unlike viscosity solutions, this method provides, by construction, a solution that enjoys the minimal regularity needed to define, in classical sense, the candidate optimal feedback map of the associated stochastic optimal control problem and, unlike the BSDEs approach, the structural restriction is here not required. On the other hand, the required smoothing property is not trivial to prove and fails to hold in many cases. However, we show that, using the generalization of -derivative that we introduce here, it is is satisfied in some important classes of problems — where instead the structural condition is not true.
Mild solutions are not regular enough to apply of Itô’s formula yet — hence, to enable to prove a verification theorem showing that the candidate optimal feedback map really provides a solution to the associated control problem. Nevertheless, they represent a first step towards this goal. Indeed, one can rely on this notion to prove that they are, in fact, strong solutions (see [39, 45]): the latter concept allows to perform the verification issue by approximation. On the other hand, at least in the case of control of Ornstein-Uhlenbeck process, already the notion of mild solution suffices to prove such a kind of result, as we will show in a subsequent paper (see also Remark 5.2 on this issue).
This paper is organized as follows. In Section 2, after the setting of notations and spaces, we introduce and study the notion of -derivative for functions between Banach spaces. This is a kind of generalized Gateaux differential, where only some directions, selected by an operator valued map , are involved. The latter notion was considered and studied in some previous papers. Precisely, it was developed in  (see also [60, Sec. 4] and ) for maps valued in the space of bounded linear operators. Here we extend this notion, fixing some features, to the case when the map is valued in the space of possibly unbounded linear operators 444This extension enables to treat a larger variety of cases (in particular the case of boundary control problems), which require to deal with unbounded control operators when the problem is reformulated in infinite dimension.. The crucial property that we prove is represented by a “pointwise" exchange property between -derivative and integration (Proposition 2.9), on which our main result relies.
Section 3 is the theoretical core of the paper. We set the notion of mild solution motivating it by an informal argument and state our main results (Theorems 3.8 and 3.10) on existence and uniqueness of solutions to the integral equation
The results are stated under the aforementioned smoothing assumption: we require that the semigroup maps continuous functions into -differentiable ones.
To show that the smoothing assumption is actually verified in several concrete circumstances, we devote Section 4 to the investigation of reasonable conditions guaranteeing the validity of it. In particular, we focus on the Ornstein-Uhlenbeck case, providing a new result that falls in the previous literature when , but extend meaningfully to other important cases when and, especially, when is unbounded. The result is contained in Theorem 4.11 and extends the known one Theorem 4.8 (contained in ). For completeness, we also report another known result (Theorem 4.17) contained in , where the smoothing assumption is verified for in the case of smooth data and invertible diffusion coefficient.
Finally, to show the implications of our results, we devote Section 5 to present a stochastic optimal control in the Hilbert space and show how the associated Hamilton-Jacobi-Bellman (HJB) equation falls, as a special case, in the class of (1.1). Then, a specific example of boundary optimal control, through Neumann type conditions, of a stochastic heat equation is provided. In this example, we discuss the validity of all the assumptions that allow to apply our main result through the use of Corollary 4.12. As far as we know, this is the first time that the HJB equation associated to this kind of problem is approached by means of solutions that have more regularity than viscosity solutions.
In this section we provide some preliminaries about spaces and notations used in the rest of the paper. Also, we provide the notion of -gradient for functions defined on Banach spaces and some properties of this object.
2.1 Spaces and notation
Here we introduce some spaces and notations.
2.1.1 General notation and terminology
If is a Banach space we denote its norm by . The weak topology on is denoted by . If is also Hilbert, we denote its inner product by . Given and , the symbol denotes the closed ball in centered at of radius . In all the notations above, we omit the subscript if the context is clear.
If a sequence , where is Banach, converges to in the norm (strong) topology we write . If it converges in the weak topology we write .
If is a Banach space, we denote by its topological dual, i.e. the space of all continuous linear functionals defined on . The operator norm in is denoted by . The duality with is denoted by . If is a Hilbert space, unless stated explicitly, we always identify its dual with through the standard Riesz identification.
All the topological spaces are intended endowed with their Borel -algebra. By measurable set (function), we always intend Borel measurable set (function).
2.1.2 Spaces of linear operators
If are Banach spaces with norm and , we denote by the set of all bounded (continuous) linear operators with norm , using for simplicity the notation when . is a Banach algebra with identity element (simply if unambiguous).
If are Banach spaces, we denote by the space of closed densely defined possibly unbounded linear operators , where denotes the domain. Clearly, . Given , we denote its adjoint operator by and its range by .
Let be a separable Hilbert space. We denote by (subset of ) the set of trace class operators, i.e. the operators such that, given an orthonormal basis of , the quantity
is finite. The latter quantity is independent of the basis chosen and defines a norm making a Banach space. The trace of an operator is denoted by , i.e. . The latter quantity is is finite and, again, independent of the basis chosen. We denote by the subset of of self-adjoint nonnegative (trace class) operators on . Note that, if , then .
If are separable Hilbert spaces, we denote by (subset of ) the space of Hilbert-Schmidt operators from to , that is the spaces of operators such that, given an orthonormal basis of , the quantity
is finite. The latter quantity is independent of the basis chosen and defines a norm making a Banach space. It is actually a Hilbert space with the scalar product
where is any orthonormal basis of . We refer to [22, App. A] for more details on trace class and Hilbert-Schmidt operators.
2.1.3 Function spaces
Let be Banach spaces and . We denote by (respectively, ) the space of measurable (respectively, measurable and bounded) functions from into . The space is a Banach space with the usual norm
We denote by (respectively, ) the space of continuous (respectively, continuous and bounded) functions from into . The space is a Banach space with the norm (2.1).
Given , we define (respectively, ) as the set of all functions (respectively, ) such that the function
belongs to (respectively, ). These spaces are made by functions that have at most polynomial growth of order and are Banach spaces when they are endowed with the norm (respectively, We will write (respectively, ) when the spaces are clear from the context. When the above spaces reduces to and and we keep this notation to refer to them.
If , the Gateaux (resp., Fréchet) derivative of at the point is denoted by (resp., ).
We define the space as the space of maps such that, for every , (555This property is usually called strong continuity of .) and the space as the space of maps such that, for every , . When , we write .
Let and let be three Banach spaces. The space ) is Banach when endowed with the norm
(When it is clear from the context, we simply write .)
First of all, we observe that the right hand side of (2.3) is finite due to a straightforward application of the Banach-Steinhaus theorem, so it clearly defines a norm.
Let be a Cauchy sequence in . Then, for each , by completeness of , in for some . On the other hand, by completeness of , we also have, for each , where in for some . By uniqueness of the limit we have for each and . Hence, . It remains to show that with respect to , that is
Now, for every with and , we have
We conclude as is a Cauchy sequence in .
2.1.4 Spaces of stochastic processes
Let be a filtered probability space satisfying the usual conditions. Given , , and a Banach space , we denote by the set of all (equivalence classes of) progressively measurable processes such that
This is a Banach space with the norm . Next, we denote by the space of all (equivalences classes of) progressively measurable processes such that for every .
Given , , and a Banach space , we denote by the space of all (equivalence classes of) progressively measurable processes admitting a version with continuous trajectories and such that
This is a Banach space with the norm . We denote by the set of all (equivalences classes of) progressively measurable processes such that for every .
2.2 Bochner integration
Let , let be a Banach space, and let be measurable. We recall that, if , then is Bochner integrable, and we write . Moreover, in this case
Finally, we recall that, if is separable, by Pettis measurability Theorem [68, Th. 1.1], is measurable if and only if is measurable for every .
Here we set and investigate the notion of -derivative for functions , where are Banach spaces. The latter notion was defined in  (see also [60, Sec. 4] and ) when is a map with Banach space. Here we extend the definition requiring only that .
Let , , be three Banach spaces and let , .
The -directional derivative at a point along the direction is defined as
assuming that this limit exists in .
We say that is -Gateaux differentiable at a point if it admits the -directional derivative along every direction and there exists a bounded linear operator such that for every and every . In this case is called the -Gateaux derivative at .
We say that is -Gateaux differentiable on if it is -Gateaux differentiable at every point and call the object the -Gateaux derivative of .
We say that is -Fréchet differentiable (or simply -differentiable) at a point if it is -Gateaux differentiable at and if the limit in (2.5) is uniform for . The latter is equivalent to
In this case, we denote and call the -Fréchet derivative (or simply the -derivative) of at .
We say that is -Fréchet differentiable on (or simply -differentiable) if it is -Fréchet differentiable at every point and call the object the -Fréchet derivative (or simply the -derivative) of .
Note that, in the definition of the -derivative, one considers only the directions in selected by the image of . This is similar to what is often done in the theory of abstract Wiener spaces, considering the -derivative, where is a subspace of , see e.g. . Similar concepts are also used in ,  and in [21, Sec.3.3.1]. A generalized notion of -derivative in spaces where separable Hilbert space and is a suitable Radon measure, is considered in relation to Dirichlet forms, see e.g.  (or also [16, Ch. 3], where it is called Malliavin derivative). In all these references is a bounded operator.
Clearly, if is Gateaux (resp., Fréchet) differentiable at , and , it turns out that is -Gateaux (resp., Fréchet) differentiable at and
i.e. the -directional derivative in direction is just the usual directional derivative at a point in the direction .
If , then the (Gâteaux or Fréchet) -derivative takes values in . Hence, if is a Hilbert space, identifying with its topological dual , we consider and write for . Similarly we do for the -Fréchet derivative.
In the same spirit, when and both and are Hilbert spaces, we identify the spaces and with their topological dual spaces and . Hence, whenever is Gateaux (resp., Fréchet) differentiable at , the identity (2.6) becomes
and similarly for the -Fréchet derivatives.
The notion of -derivative allows to deal with functions which are not Gateaux differentiable, as shown by the following example.
Let be defined by . Clearly, does not admit directional derivative in the direction at the point . On the other hand, if we consider , defined by , where , then admits -Fréchet derivative at every .
When only takes values in , that is when we deal with the possibility that is unbounded, then even if is Fréchet differentiable at all points , the -Fréchet derivative may not exist in some points. Indeed, consider the following example.
Let be Hilbert spaces, let be a closed densely defined unbounded linear operator on , and let be its (unbounded) adjoint. Next, let defined by and let be defined by . Clearly, is Fréchet differentiable at every and . Then, by definition of -directional derivative, we have
On the other hand, if was -Fréchet differentiable at every , we should have for every , and therefore
It should follow , which is not the case if is any genuinely unbounded linear operator.
Let , , be Banach spaces, let , and let . We define the spaces of functions
When we use the notation and . Moreover, when we omit it in the notation.
Definition 2.6 (Pseudoinverse).
Let be a uniformly convex Banach space, let be a Banach space, and let . The pseudoinverse of is the linear operator defined on and valued in associating to each the (unique) element in with minimum norm in . 666Existence and uniqueness of such an element follows from the fact that is a closed operator and applying the results of [25, Sec. II.4.29, p. 74]).
If in Definition 2.6 is Hilbert space, then
Now we deal with the possibility of performing the -differentiation under the integral sign in pointwise and functional sense. Due to the integrability issues clarified at the beginning of Section 3, we will make use only of the pointwise exchange property (Proposition 2.9). However, since the analogue functional property has not been well developed in the literature, we establish the result also in this case (Corollary 2.12) providing a complete proof.
and are Hilbert spaces. The map is such that
for every ; we denote by the common domain;
for every ; we denote by the common range;
Let be the pseudo-inverse of according to Definition 2.6. The map is locally bounded for every .
Let Assumption 2.8 hold. Let and let be measurable and such that
(resp., ) for a.e. ;
there exists such that, for a.e. and every ,
Then, the function , is well defined and belongs to (resp., ) and
First of all, note that the fact that is well defined and belongs to follows from (i) and (2.7) by dominated convergence.
We prove now the other claims for the -Gateaux gradient; the proof for the -Fréchet gradient is obtained just replacing by and by in the following. Let , . Set
Then, using Assumption 2.8(i)-(ii), we can write
Then, arguing as in the standard one dimensional case, dominated convergence combined with a generalization of Lagrange Theorem to -valued functions (see [73, Prop. 3.5, p. 76]) yields
This also shows that
Assumption 2.8 holds true with locally bounded replaced by continuous in (iii).
We give the proof for . The proof is analogous.
Let be a Cauchy sequence in . In particular, is a Cauchy sequence in , so that converges to a function .
Now, for all , is a Cauchy sequence of linear bounded operators in , so that converges to a linear bounded operator . On the other hand, for all , the sequence is a Cauchy sequence in so that converges to a function . Hence, we have , which yields .
Now, note that, by definition of , we have whenever . It follows
We are going prove that and . Let and . Set, for , and Then, arguing as in the proof of Proposition 2.9, we get
As , by the Banach-Steinhaus Theorem the family is a family of uniformly bounded operators in . Setting
we have, for every ,
Now, as , we have the convergences
So, from 2.13, we get