Regularization and Propagation in a Hamilton–Jacobi–Bellman-Type Equation in Infinite-Dimensional Hilbert Space

Abstract

This paper is devoted to new propagation and regularity results for a class of first-order Hamilton–Jacobi–Bellman-type problems in a separable infinite-dimensional Hilbert space. Specifically, the related Cauchy problem is investigated by employing the Faedo–Galerkin approximation method. Under some structural assumptions, the main result is obtained by using the probabilistic representation formula of the solution in order to define the weak continuity assumptions, by assuming the existence of a symmetric positive definite Hilbert–Schmidt operator and by employing modulus continuity arguments.

1. Introduction

The mathematical modeling of complex systems requires great attention considering that it usually is based on a system of nonlinear differential equations coupled to specific initial and/or boundary conditions. Different initial/boundary value problems can be defined depending on the form of the boundary conditions (Dirichlet, Neumann, Robin [1,2,3]). The well-posedness of the initial/boundary value problem is at the core of the mathematical analysis; the existence and uniqueness of solutions and the continuous dependence on the initial data are usually called the Hadamard conditions [4,5]. However, from the mathematical analysis viewpoint, the regularization and the propagation of some properties are important areas that have been recently gained much attention, particularly in infinite-dimensional spaces, where the definition of some operators and operations needs to be highlighted. In this context, a general theory is missing, and some methods derived from nonlinear analysis have been proposed; see, among others, semigroup theory and stochastic analysis theory [6,7].

Mathematical problems based on Hamilton–Jacobi–Bellman-type equations in infinite-dimensional Hilbert spaces have been investigated in [8,9,10]. A seminal paper on the study of the convexity of a Hamilton–Jacobi–Bellman-based problem is [11]; viscosity solution theory in an infinite-dimensional space has been also proposed in [12] and further developments have been investigated in [13,14]. Mild solutions based on stochastic representation formulas have recently attracted much attention [15]. Deep neural network approximations [16] and the relationships with the incompressible Navier–Stokes equations [17] have also been investigated. Viscosity solutions to the Hamilton–Jacobi–Bellman equations associated with sublinear Lévy-type processes have been studied in [18]. Readers interested in the numerical approximation of Hamilton–Jacobi–Bellman equations is referred to [19].

A particular Hamilton–Jacobi–Bellman mathematical problem is based on the heat equation with the source term

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{\partial u}{\partial t}-\frac{1}{2}\Delta u=f\left(x\right),t>0,x\in H}\hfill \\ \hfill & {\displaystyle u(x,0)=u_0\left(x\right),x\in H}\hfill \end{array}\right.$$

(1)

where H denotes a real separable infinite-dimensional Hilbert space. Recently, problem (1) has been mathematically analyzed in order to gain some regularization and propagation properties [20].

This paper aims at generalizing the investigations of the paper [20] to the following nonlinear problem based on a Hamilton–Jacobi–Bellman-type equation:

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{\partial u}{\partial t}}-{\displaystyle \frac{1}{2}}\Delta u+F(x,\nabla u)=0,t>0,x\in H\hfill \\ \hfill & u(x,0)=u_0\left(x\right),x\in H.\hfill \end{array}\right.$$

(2)

A specific example of (2) is defined by the following equation:

$${\displaystyle \frac{\partial u}{\partial t}}-{\displaystyle \frac{1}{2}}\mathrm{trace}\left(A_0{D}^{2}u\right)+F(x,Du)=0,$$

(3)

where $A_0$ is a linear, bounded, positive, self-adjoint, nuclear operator on H (or $A_0$ is a degenerate symmetric matrix such that $A_0⩾0$), and F a time-independent function called a Hamiltonian. Readers interested in some mathematical problems based on Equation (3) are referred to papers [21,22,23] and the references cited therein.

This paper deals with the mathematical analysis of problem (2) by employing the Faedo–Galerkin approximation method consisting of introducing finite-dimensional approximations of Equation (2); see, among others, paper [24]. Moreover, the probabilistic representation formula of the solution is used in order to define the weak continuity assumptions [25]. The main technical assumptions are based on the existence of a symmetric positive definite Hilbert–Schmidt operator B and on the B-modulus continuity arguments [26].

The contents of the present paper are outlined as follows: Section 2 is devoted to the notations and background; Section 3 contains the main assumptions and the statement of the main result; and Section 4 and Section 5 are devoted to the main elements of the proof of the main result.

2. Background and Notations

In what follows, $H=(H,⟨·,·⟩,\parallel ·\parallel )$ denotes a real separable Hilbert space and $x=\sum x_i{e}_i$ its generic point. According to the Faedo–Galerkin method (see [27]), the properties in the space H can be studied by means of the introduction of the vectorial space $H_N=\mathrm{Vect}(e_1,\dots ,e_N)$. Let $P_N$ be the orthogonal projection on $H_N$ and $Q_N=\mathbf{1}-P_N$ the orthogonal projection on the orthogonal space of $H_N$. The spaces $H_N$ and ${\mathbb{R}}^N$ are isomorphic spaces.

Let $T>0$ and $\phi :(0,T)\times H\to \mathbb{R}$ be a real value function; $D\phi$ and $D^2\phi$ denote the first- and second-order Fréchet derivatives of $\phi$ with respect to the “space” variable $x\in H$, respectively.

Let $\mathcal{S}\left(N\right)$ be the set of symmetric $N\times N$ matrices and A a symmetric matrix, e.g., $D^2\phi$. By definition, $A\ge a$ means that $\sigma \left(A\right)\subseteq [a,\infty )$. The 2-norm of A is defined as follows:

$${\left|A\right|}_2=\underset{\left|x\right|=1}{sup}\left|Ax\right|=\underset{\left|x\right|=1}{sup}{(A^{2}x,x)}^{1/2},$$

and, in particular, if $A⩾0$, one has

$${\left|A\right|}_2={\lambda }_1{\left(A^2\right)}^{1/2}={\lambda }_1\left(\right|A\left|\right)={\lambda }_1\left(A\right),$$

where ${\lambda }_1$ denotes the largest eigenvalue of $A^2$.

Let $\Omega$ be an open set of H. The following functional spaces will be employed in the present paper:

• $C(\Omega )=\{u:\Omega ⟼\mathbb{R};u \mathrm{i}\mathrm{s} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}\}$;
• $C_b(\Omega )=\{u:\Omega ⟼\mathbb{R};u \mathrm{i}\mathrm{s} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d} \mathrm{o}\mathrm{n} \mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d} \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{s} \mathrm{o}\mathrm{f} \Omega \}$;
• $UC(\Omega )=\{u:\Omega ⟼\mathbb{R};u \mathrm{i}\mathrm{s} \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{l}\mathrm{y} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}\}$;
• $BUC(\Omega )=\{u\in UC(\Omega );u \mathrm{i}\mathrm{s} \mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d}\}$;
• $C^{k,\alpha }\left(H\right)$ denotes the Hölder ($0<\alpha <1$)/Lipschitz ($\alpha =1$) space, where k is a nonnegative integer referring to $D^k\phi$.
  In particular, $C_b^{1,1}\left(H\right)=\{v\in C_b^1\left(H\right)$, Dv is Lipschitz on $H\}$.

Let B be a bounded linear operator in H and $u:H\to \mathbb{R}$; u is said to be B-continuous if $u\left(x_n\right)\to u\left(x\right)$ whenever $x_n\rightharpoonup x$ weakly in H and $B{x}_n\to Bx$ in H.

The notations $x_n\to x$ and $x_n\rightharpoonup x$ denote the norm convergence and the weak convergence of $x_n$ to x, respectively.

It is well known that

• $C_b^{1,1}\left(H\right)$ is dense in $C_b\left(H\right)$ (see [28,29]);
• If X is a finite-dimensional space, then the space $C_b^k\left(X\right)$, $k\in \{1,2,\dots \infty \}$ is dense in $C_b\left(X\right)$;
• If X is a infinite-dimensional, on the contrary, the density of $C_b^k\left(X\right)$ in $C_b^0\left(X\right)$ fails to be true for $k⩾2$ (see [29]).

3. Main Assumptions and Statement of the Main Result

This section deals with the main result of the present paper, related to the following Cauchy problem based on the Hamilton–Jacobi–Bellman-type equation:

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{\partial u}{\partial t}}-{\displaystyle \frac{1}{2}}\Delta u+F(x,\nabla u)=0,t>0,x\in H\hfill \\ \hfill & u(x,0)=u_0\left(x\right),x\in H\hfill \end{array}\right.$$

(4)

where H denotes a real separable infinite-dimensional Hilbert space.

A time-independent Hamiltonian F is taken into account. It is worth stressing that the notation $\nabla u$ instead of $Du$ is employed to remark that u is a scalar function.

For convenience, a specific case of $F(x,\nabla u)$ is considered, namely $F(x,\nabla u)\equiv F(\nabla u)+f\left(x\right)$. Accordingly, the Cauchy problem (4) reads

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{\partial u}{\partial t}}-{\displaystyle \frac{1}{2}}\Delta u+F(\nabla u)=f\left(x\right),t>0,x\in H\hfill \\ \hfill & u(x,0)=u_0\left(x\right),x\in H\hfill \end{array}\right.$$

(5)

where $F:H\to \mathbb{R}$ is a given measurable function belonging to $C^{\infty }\left(C^{1,1}\left(H\right)\right)$.

The following problem is now discussed in order to clarify to the reader the method of the subsequent sections. The well-known Eikonal-type problem reads

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{1}{2}}{|\nabla u|}^2=0,\hfill \\ \hfill & u(x,0)=u_0\left(x\right)\in BUC\left(H\right).\hfill \end{array}\right.$$

(6)

In the original approach [30,31], the following approximated equation is considered:

$${\displaystyle \frac{\partial u^{\epsilon }}{\partial t}}-{\displaystyle \frac{\epsilon }{2}}\Delta u^{\epsilon }+{\displaystyle \frac{1}{2}}{|\nabla u^{\epsilon }|}^2=0,u^{\epsilon }(x,0)=u_0\left(x\right),$$

(7)

for $\epsilon >0$. The operator ${\displaystyle \frac{\epsilon }{2}}\Delta$ in (7) regularizes the Eikonal equation, known as the vanishing viscosity method. The aim is to let $\epsilon \downarrow 0$ and to study the limit of the family ${\left(u^{\epsilon }\right)}_{\epsilon >0}$.

Usually, ${\left(u^{\epsilon }\right)}_{\epsilon >0}$ is bounded and locally equicontinuous on $H\times (0,T)$. By using the Arzela–Ascoli theorem [32], one deduces that

$$u^{{\epsilon }_j}⟶u\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y} \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{l}\mathrm{y} \mathrm{i}\mathrm{n} H\times (0,T),$$

for some subsequence $u^{{\epsilon }_j}$ and some limit function $u\in C(H\times (0,T\left)\right)$.

It seems that u is a type of solution of (6); however, u is only continuous and no information about $\nabla u$ and $u_t$ is available. Moreover, (6) is fully nonlinear and does not have a divergence structure; thus, no integration by parts or weak convergence techniques can be employed in order to justify that u is a weak solution. Then, the maximum principle is used to obtain the notion of a weak solution (viscosity solution).

The partial differential equation of problem (6) shares a property with the heat equation. Indeed, straightforward computation shows that if $u^{\epsilon }$ is a smooth solution of the above equation, then the following Hopf–Cole transform [33]

$$\phi =e^{-u/\epsilon },$$

(8)

satisfies the following linear heat problem:

$${\phi }_t-{\displaystyle \frac{\epsilon }{2}}{\Delta \phi =0,\phi |}_{t=0}={\phi }_0=e^{-u_0/\epsilon }.$$

(9)

If $\epsilon =1$, by classical linear theory, the solution $\phi$ of (9) admits the following integral representation:

$$\phi (x,t)={\left(4\pi t\right)}^{-{\displaystyle \frac{N}{2}}}{\int }_{{\mathbb{R}}^N}{e}^{-{\displaystyle \frac{{|x-y]}^2}{4t}}}{e}^{-{\displaystyle \frac{u_0\left(x\right)}{2}}}dy,$$

(10)

and, hence, for $u_0\in BUC\left(H\right)$ and ${\phi }_0=e^{-u_0}$, and by inverting the Hopf–Cole transform (since it is Lipschitz between $u_0$ and $\phi$), we obtain

$$u(x,t)=-2\mathrm{l}\mathrm{o}\mathrm{g}\left({\left(4\pi t\right)}^{-{\displaystyle \frac{N}{2}}}{\int }_{{\mathbb{R}}^N}{e}^{-{\displaystyle \frac{{|x-y]}^2}{4t}}}{e}^{-{\displaystyle \frac{u_0\left(x\right)}{2}}}dy\right),$$

(11)

which is solution of the quasilinear problem (6). Consequently, transformation (8) transforms the nonlinear Hamilton–Jacobi–Bellman-type equations into linear evolution equations.

The mathematical analysis performed in the present paper requires the following structural assumptions on $u_0$ and f.

(H1) 

There exists a symmetric positive definite Hilbert–Schmidt operator B such that $B=B^{*}>0$ (the adjoint operator of B) and we assume $\omega$ (a modulus of continuity) to be continuous, increasing and subadditive $\omega \left(r\right)\to 0$ as $r\downarrow 0$ on $[0,+\infty ]$ such that $u_0\in C_b\left(H\right)$ and

$$|u_0\left(x\right)-u_0\left(y\right)|⩽\omega (\left|B(x-y)\right|),$$

(12)

$$\left|f\right(x)-f(y\left)\right|⩽\omega \left(\right|B(x-y)\left|\right).$$

(13)

The following further assumption is introduced (first regime).

(H2) 

There exists a modulus of continuity $\omega :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that

$$|{\phi }_0\left(x\right)-{\phi }_0\left(y\right)|⩽\omega (|Bx-By|)$$

(14)

Inequality (14) is propagated for the heat equation

$$|u_0\left(x\right)-u_0\left(y\right)|⩽\tilde{\omega }\left(\right|Bx-By\left|\right)$$

(15)

where

$${\displaystyle \frac{1}{C_0}}\omega ⩽\tilde{\omega }⩽C_0\omega ,$$

where $C_0$ depends only on the norm of the supremum of $u_0$.

The third assumption that we introduce is the following (second regime).

(H3) 

The bounded linear operator B is such that

$$|B^{-1}\nabla u|⩽C_0$$

(16)

which means that $\omega \left(z\right)=C_{0}z$; then, it propagates and it occurs indifferently from ${\phi }_0$ to $u_0$.

It may be useful to remark that hypothesis (16) is equivalent to $u⟺\phi$ (therefore, whether for u or for $\phi$, the Cole–Hopf transformation leaves it invariant).

Bearing all of the above in mind, we deduce that, in particular, $\Delta {\phi }_0$ is bounded (however, $\Delta {\phi }_0$ is not $\Delta u_0$, but we have $\Delta u_0$ and ${|\nabla u|}^2$ is bounded). We are in the second regime, i.e.,

$$\left\{\begin{array}{cc}\hfill & {\displaystyle |B^{-1}\nabla u|⩽C_0}\hfill \\ \hfill & {\displaystyle \omega \left(z\right)=C_{0}z}\hfill \end{array}\right.$$

(17)

is propagated; the second regime establishes that as soon as the $u_0$ verifies hypothesis (16) or (17), then

-

the Laplacian and ${\displaystyle {|\nabla u|}^2}$ have a meaning;

-

u satisfies the Lipschitz condition for positive times.

The third regime is rewritten as follows.

(H4) 

The operator B is such that

$$|B^{-1}{D}^2\phi B^{-1}|⩽C_0.$$

(18)

On the other hand, we have

$$|B^{-1}{D}^2\phi B^{-1}|=e^{-u}\left|B^{-1}(D^{2}u-Du\otimes \Delta u)B^{-1}\right|,$$

(19)

from which we deduce the term that interests us, i.e., $B^{-1}{D}^2\phi B^{-1}$, but we have an additional term, which is $(B^{-1}Du\otimes B^{-1}\Delta u)$:

$$B^{-1}(D^{2}u-Du\otimes \Delta u)B^{-1}=B^{-1}{D}^{2}u{B}^{-1}-(B^{-1}Du\otimes B^{-1}Du).$$

Accordingly, the last assumption and the third regime imply the second regime.

By adapting the argument, we deduce that hypothesis (18) entails hypothesis (16) and therefore $(B^{-1}Du\otimes B^{-1}\Delta u)$ is automatically bounded using the boundedness (18).

The following main result holds true.

Theorem 1.

Let $H_N$ be a sequence of increasing vector subspaces of dimension N that converges to H. Let $u_N$ be a solution of the following problem:

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{\partial u_N}{\partial t}-\Delta u_N+F\left(D{u}_N\right)=f\left(x\right),(x,t)\in H_N\times (0,\infty )}\hfill \\ \hfill & {\displaystyle u_N(x,0)=u_0\left(x\right),}\hfill \end{array}\right.$$

(20)

such that

$$|u_N(x,t)-u(x,s)|⩽m(t-s),\forall t⩾s,\forall N,\forall x,$$

(21)

where m is the modulus of continuity in t, namely $m\left(0\right)=0$. Then,

(1)

$u_N$ converges uniformly in $(x,t)$ on every bounded set of H to a function $u\in BUC(H\times [0,\infty \left)\right)$:

$$|u(P_{N}x,t)-u_N(x,t)|\underset{N\to \infty }{\to }0$$

(22)

The function u is such that

(i)

$$\left|u\right(x,t)-u(y,t\left)\right|⩽\omega \left(\right|B(x-y)\left|\right).$$

(23)

(ii)

Since $u_0\left(x\right)=v_0\left(Bx\right)$, we have

$$u(x,t)=v(Bx,t),wherev\in BUC(H\times [0,\infty \left)\right),$$

(24)

$v_0,v$ are uniformly continuous for the norm of H on a vector subspace $\mathtt{BH}$, which is the image of H by B in H, which allows us to give a meaning to $u_0(x+W_t)$ and defines

$$u_0(x+W_t)=v(Bx+B{W}_t,t),B{W}_t\in H,$$

(25)

so that

$$u(x,t)=\mathbb{E}\left[u_0(x+W_t)\right].$$

(26)

(2)

Moreover, if

$$\omega \left(z\right)=Lz,L>0,$$

then

$$Tr|D^{2}u|<\infty and uniformly sommable with respect to x,t$$

and u is Lipschitzian with respect to t.

(3)

If $u_0$ satisfies

$$|B^{-1}\nabla u\left(x\right)-B^{-1}\nabla u\left(y\right)|⩽C_0{\left|B(x-y)\right|}_H$$

(27)

then $u\left(t\right)$ for all $t⩾0$ satisfies (27).

The proof of Theorem 1 follows in the arguments of the following sections.

4. Formal a Priori Estimations

This section investigates the viscosity solution method for the problem

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{\partial u}{\partial t}-\frac{1}{2}\Delta u=f\left(x\right)-F\left(Du\right)(x,t)\in H\times (0,\infty )}\hfill \\ \hfill & {\displaystyle u(x,0)=u_0\left(x\right)\in BUC\left(H\right).}\hfill \end{array}\right.$$

(28)

Firstly, an a priori estimate is investigated. The investigations focus on the $C^{1,1}$ regularity and not on the existence and uniqueness of a viscosity solution result. Most of the necessary proofs are sketched in order to ensure that this part is self-contained.

The following further assumption is required.

(H5) 

Assume that

(i)

F is B-continuous;

(ii)

$u_0$ and f are Lipschitzian bounded functions on H.

The following lemma holds true.

Lemma 1.

Le u be the solution of (28) that satisfies hypotheses (H5). Then,

(i)

The solution u is B-continuous on $[0,T]$;

(ii)

The following estimate holds

$$|B^{-1}\nabla u_0|⩽C_0⟹|B^{-1}\nabla u|⩽Con[0,T];$$

(29)

(iii)

If $u_0\in C^{1,1}$, then

$${\parallel u\left(t\right)\parallel }_{C^{1,1}}⩽C_{T}on[0,T];$$

(30)

(iv)

Furthermore,

$${\parallel u\left(t\right)\parallel }_{C^{1,1}}⩽{\displaystyle \frac{C_T}{\sqrt{t}}}on[0,T];$$

(31)

(v)

Moreover, in general,

$$|B^{-1}{D}^{2}u\left(t\right)B^{-1}|⩽C_{T}on[0,T].$$

(32)

Proof of Lemma 1. 

The probabilistic representation of the solution of (28) is given by

$$u(x,t)=\mathbb{E}[u_0(x+W_t)+{\int }_0^t[f-F\left(Du\right)](t-s)ds],$$

(33)

where $\mathbb{E}$ denotes the expectation in the probability space $(\Omega ,\mathcal{F},P)$ and $W_t$ represents the H-valued Brownian motion. □

Differentiating (33) with respect to $\xi$ and using the analytical expression of the Gaussian density of $W_t$, one has [34,35]

$$u_{\xi }(x,t)=\mathbb{E}[u_0(x+W_t){\displaystyle \frac{W_t·\xi }{t}}+{\int }_0^t[f-F\left(Du\right)](t-s){\displaystyle \frac{W_s·\xi }{s}}ds],\left|\xi \right|=1.$$

(34)

By differentiating (34) twice with respect to $\xi$, one has

$$u_{\xi \xi }(x,t)=\mathbb{E}\left[{\partial }_{\xi }{u}_0(x+W_t){\displaystyle \frac{W_t·\xi }{t}}+{\int }_0^t{\partial }_{\xi }[f-F\left(Du\right)](t-s){\displaystyle \frac{W_s·\xi }{s}}ds\right].$$

(35)

We know that $\nabla u$ is bounded; ${\displaystyle \frac{W_t·\xi }{t}}$ increases by $|W_t|$ in the direction $\xi$ as it is divided by t and it increases with $t\sqrt{t}$. Likewise, the same is true for ${\partial }_{\xi }f$. Then, necessarily, we have that

$$|u_{\xi \xi }(x,t)|⩽{\displaystyle \frac{C}{\sqrt{t}}}+C{\int }_0^t{\displaystyle \frac{\sqrt{s}}{s}}ds+C_0{\int }_0^t{\left|D^{2}u\right|}_2(t-s){\displaystyle \frac{ds}{\sqrt{s}}}.$$

(36)

Therefore, $D^{2}u\left(t\right)$ is dominated by

$$|D^2{u\left(t\right)|}_2⩽{\displaystyle \frac{C}{\sqrt{t}}}+C\sqrt{t}+C_0{\int }_0^t{\left|D^{2}u\right|}_2(t-s){\displaystyle \frac{ds}{\sqrt{s}}},$$

(37)

where

$$|D^2{u}_N{|}_2=\underset{\left|\xi \right|=\left|\eta \right|=1}{sup}{D}^2{u}_N(\xi ,\eta ).$$

(38)

If $u_0$ belongs to $C^{1,1}$, we can simply replace ${\displaystyle \frac{{\displaystyle C}}{{\displaystyle \sqrt{t}}}}$ with C; then, we would not even have a singularity. Inequality (37) is a type of Gronwall inequality, but, as ${\displaystyle \frac{1}{\sqrt{s}}}$ is summable, this is enough. We denote

$$M\left(t\right)=e^{-\lambda t}\left|D^{2}u\right|$$

(39)

where $\lambda$ is unknown.

It is easy to observe that

$${\int }_0^t|D^2{u|}_2(t-s){\displaystyle \frac{1}{\sqrt{s}}}ds={\int }_0^t|e^{-\lambda (t-s)}{\left|D^{2}u(t-s)\right|}_2{e}^{-\lambda s}{\displaystyle \frac{ds}{\sqrt{s}}}.$$

(40)

The foregoing estimate is useful, since the left-hand side of (40) is dominated by

$${\int }_0^t|e^{-\lambda (t-s)}{\left|D^{2}u(t-s)\right|}_2{e}^{-\lambda s}{\displaystyle \frac{ds}{\sqrt{s}}}⩽\underset{[0,t]}{sup}M\left(s\right){\int }_0^t{\displaystyle \frac{e^{-\lambda s}}{\sqrt{s}}}ds.$$

(41)

Since ${\displaystyle {\int }_0^t\frac{e^{-\lambda s}}{\sqrt{s}}ds}$ is a constant on $\lambda$, then

$${\int }_0^t{\displaystyle \frac{e^{-\lambda s}}{\sqrt{s}}}ds⩽{\int }_0^{\infty }{\displaystyle \frac{e^{-\lambda s}}{\sqrt{s}}}ds.$$

(42)

By setting $\lambda s=x^2$, elementary computation gives

$${\int }_0^{\infty }{\displaystyle \frac{e^{-\lambda s}}{\sqrt{s}}}ds={\int }_0^{\infty }{e}^{-x^2}{\displaystyle \frac{2x\sqrt{\lambda }}{x\lambda }}dx={\displaystyle \frac{C_0}{\sqrt{\lambda }}}$$

(43)

and, bearing (43) in mind, we deduce from (41) that

$$\underset{[0,t]}{sup}M\left(s\right){\int }_0^t{\displaystyle \frac{e^{-\lambda s}}{\sqrt{s}}}ds=\underset{[0,t]}{sup}M\left(s\right){\displaystyle \frac{C_0}{\sqrt{\lambda }}}.$$

(44)

The easiest consequence of (44) is that, for every $u_0\in C^{1,1}$, we obtain

$$|D^2{u\left(t\right)|}_2⩽C+{\displaystyle \frac{C_0}{\sqrt{\lambda }}}\underset{[0,T]}{sup}M\left(s\right).$$

(45)

Thus, ${\displaystyle \frac{C_0}{\sqrt{\lambda }}}$ is less than 1 provided that we choose $\lambda$ to be large enough and then (36) is satisfied.

We finally reach the conclusion that if $u_0\in C^{1,1}$, then the solution is $C^{1,1}$.

We have proven that if $u_0\in C^{1,1}$ and is bounded, then

$$|D^2{u\left(t\right)|}_2⩽{\displaystyle \frac{C_0}{\sqrt{t}}}$$

(46)

$${\parallel u\left(t\right)\parallel }_{C^{1,1}}⩽C_{T}on[0,T]$$

(47)

and, in general,

$${\parallel u\left(t\right)\parallel }_{C^{1,1}}⩽{\displaystyle \frac{C_T}{\sqrt{t}}}on[0,T].$$

(48)

Bearing all of the above in mind, the following estimation for $D^{2}u$ on $[0,T]$, with a fixed T, holds true.

Claim 1. 

Let $u_0$ be a Lipschitzian bounded function on H. Then, for $t\in [0,T]$, a subsolution is such that

$${\displaystyle |D^2{u\left(t\right)|}_2⩽\frac{C_0}{\sqrt{t}}+C{\int }_0^t\frac{\parallel |D^{2}u{{|}_2\parallel }_{\infty }\left(s\right)}{\sqrt{t-s}}ds.}$$

(49)

Proof of Claim 1. 

To prove the integro-differential inequality, we consider a problem with eigenvalues and we consider two different cases: $t⩽T_0$ and $T_0<t<2T_0$.

• The case $t⩽T_0$. On the one hand, the second term on the right-hand side of (49) is an integrable kernel; by placing it on the left-hand side, we have a jump operator that verifies the principle of the maximum up to the first eigenvalue and therefore there exists a solution.

On the other hand, note that the total mass of this kernel is equal to $\sqrt{T}$, which is the worst case that can be obtained over the whole interval $[0,t]$, so that if $C\sqrt{T}<1$, we are not sure whether we have crossed the first eigenvalue. This is equivalent to

$$\parallel |D^{2}u\left(t\right){{|}_2\parallel }_{\infty }⩽{\displaystyle \frac{K}{\sqrt{t}}}.$$

(50)

To this end, we look for a supersolution in the form

$$u(x,t)={\displaystyle \frac{K}{\sqrt{t}}},K>C_0.$$

(51)

In order to ensure that $u(x,t)$ is a supersolution, we need to evaluate the second term of (49). Using (50), we thus find

$$C{\int }_0^t{\displaystyle \frac{\parallel |D^{2}u{{|}_2\parallel }_{\infty }\left(s\right)}{\sqrt{t-s}}}ds⩽CK{\int }_0^t{\displaystyle \frac{ds}{\sqrt{s}\sqrt{t-s}}}ds.$$

(52)

With the change in variable $s=tu$, the right-hand side of (52) is

$$CK{\int }_0^t{\displaystyle \frac{1}{\sqrt{s}\sqrt{t-s}}}ds=CK{\int }_0^1{\displaystyle \frac{du}{\sqrt{u}\sqrt{1-u}}}.$$

(53)

Replacing this in (49), this entails that

$$|D^2{u\left(t\right)|}_2⩽{\displaystyle \frac{C_0}{\sqrt{t}}}+C_{1}K,$$

(54)

and, therefore, again, we do not obtain a global subsolution, but, for $t⩽T_0$, we have a subsolution of the form $K/\sqrt{t}$.

• For $t>T_0$, it suffices to note that, using the estimate (49), we have a bound that, after any time, does not explode (we are far from 0, and there is no more explosion), and, since the initial condition is not $C^{1,1}$, we obtain $1/\sqrt{t}$. □

Subsolution of (28). We proceed by contradiction using an argument of the maximum principle. Without loss of generality, we assume that the density $u_0\in C^{1,1}$.

Then, for all K and t that are small enough, one has

$$\parallel |D^{2}u\left(t\right){{|}_2\parallel }_{\infty }<{\displaystyle \frac{K}{\sqrt{t}}}$$

(55)

provided that we choose t to be small enough. Therefore, for $t<T_0$, we can assume that

$${\displaystyle \frac{C_0}{\sqrt{t}}}+C_{1}K<{\displaystyle \frac{K}{\sqrt{t}}}\mathrm{on}[0,T_0],$$

(56)

which means that $K/\sqrt{t}$ is a subsolution. Moreover, we wish to be sure that $D^{2}u<K/\sqrt{t}$; this is certainly the case provided that we choose t to be small enough, and we will examine the first time that it occurs with $K/\sqrt{t}$, if this first time exists.

Let $\tau$ be defined as follows:

$$\tau =inf(t⩾0,t⩽T_0,\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\parallel |D^{2}u\left(t\right){{|}_2\parallel }_{\infty }={\displaystyle \frac{K}{\sqrt{t}}})<T_0.$$

(57)

Then, we must have

$${\displaystyle \frac{K}{\sqrt{\tau }}}⩽{\displaystyle \frac{C_0}{\sqrt{\tau }}}+C{\int }_0^{\tau }{\displaystyle \frac{\parallel |D^{2}u\left(t\right){{|}_2\parallel }_{\infty }\left(s\right)}{\sqrt{t-s}}}ds.$$

(58)

Since, for $[0,\tau ]$, we have less than $K/\tau$, by definition of this first time, this entails that

$${\displaystyle \frac{\parallel |D^{2}u\left(t\right){{|}_2\parallel }_{\infty }\left(s\right)}{\sqrt{t-s}}}⩽{\displaystyle \frac{K}{\sqrt{\tau }}}{\displaystyle \frac{1}{\sqrt{\tau -s}}}.$$

Therefore,

$${\displaystyle \frac{K}{\sqrt{\tau }}}⩽{\displaystyle \frac{C_0}{\sqrt{\tau }}}+C_{1}K.$$

(59)

On the one hand, with (57), we have the equality by definition of time, and, on the other hand, by (59), we have the strict inequality. Therefore,

$${\displaystyle \frac{K}{\sqrt{\tau }}}<{\displaystyle \frac{C_0}{\sqrt{\tau }}}+C_{1}K.$$

(60)

However, we have chosen $T_0$ such that this is not possible, and we reach a contradiction. Then, the fact that it is a supersolution (even a strict one is not necessary) ensures that we cannot use $K/\sqrt{t}$. This simple argument justifies the desired result.

Obtaining a $C^{\infty }$-bound. We start with $t_0>0$ as small as we wish, but this is our new clock. At this moment, we start with $u\left(t_0\right)\in C^{1,1}$. We will write the equation satisfied by the second derivatives and use the the integration by parts formula to obtain a bound on the third derivative, which will bring us again to $1/\sqrt{t}$. More precisely, we have

$${\left(D^{2}u\right)}_t-\Delta \left(D^{2}u\right)=D^2\{F\left(Du\right)-f\}.$$

(61)

Note that the term $D^{2}f$ is bounded. The worst term in (61) is $F^{\prime }\left(Du\right)D^{3}u$. According to the previous argument, we first remark that $D^{3}u$ is dominated by

$$\parallel D^{3}u\parallel ⩽{\displaystyle \frac{C}{\sqrt{t-t_0}}}+C{\int }_{t_0}^t{\displaystyle \frac{1}{\sqrt{t-s}}}\left|D^{3}u\left(s\right)\right|ds,t>t_0.$$

(62)

In other words, the fact that we have a first-order term in the PDE (28) implies that, when we estimate the nth derivatives, we have a formula like (62) for the nth derivative. Accordingly, we have the same order derivative on the right-hand side, and this explains why we can, at this time, propagate all of the bounds.

It is interesting to obtain the propagation properties of various $C^{1,1}$-bounds which are not easy to realize. In fact, the difficulty lies in obtaining the PDE from $u=v\left(Bx\right)$, which is a “degenerate equation”, since $\mathrm{Tr}\left(D^{2}u\right)$ is monotonically increasing on $\mathcal{S}\left(N\right)$.

The most difficult aspect of the proof is to gain the propagation of the $C^2$-bounds so that we can justify our a priori estimates. To proceed, we will prove the following claim.

Claim 2.

$$\begin{array}{c}|B^{-1}{D}^2{u}_0{B}^{-1}|⩽C_0\hfill \\ |B^{-1}{D}^{2}f{B}^{-1}|⩽C_0\hfill \end{array}$$

lead to

$$|B^{-1}{D}^{2}u\left(t\right)B^{-1}|⩽C_T\mathrm{sur}[0,T].$$

(63)

Proof of Claim 2. 

A key point of the proof is to notice that there is no regularization in a high-dimensional space. First of all, we observe that

$$u(x,t)=v(Bx,t),v\in BUC(H\times [0,T\left]\right),$$

(64)

and inserting it into the PDE (28), bearing in mind that $A=B^2$, leads to the following equation:

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{\partial v}{\partial t}-Tr\left(A{D}^{2}v\right)+F\left(BDv\right)=f\left(Bx\right)}\hfill \\ \hfill & {\displaystyle v(x,0)=v_0\left(x\right)}\hfill \end{array}\right.$$

(65)

where we assume that $v_0\left(x\right)$ is Lipschitzian bounded on H. The estimate (63) implies that v is Lipschitz and the fact that $u\in C^{1,1}$ on H implies

$$|B{D}^{2}vB|⩽C.$$

(66)

Moreover, exploiting the fact that $D^{2}v$ is bounded and that $|D^2{v}_0|$ is dominated by a non-negative constant $C_0$, if we are able to prove that

$$|D^{2}v|⩽C_T,$$

(67)

then we will have the a priori estimate that we need. Within this scope, as in a finite dimension, let us fix a point $(x_0,t)$. Let $e_1$ be the eigenvector associated with the largest eigenvalue of the operator $D^{2}v$ (recall that $D^{2}v$ has a finite trace). The largest eigenvalue (in terms of the absolute value) of $|D^{2}v|$ is positive or negative. Let us denote by $v_{11}$ the second derivative in the direction $e_1$, which is assumed to be strictly positive for the sake of simplicity.

Let $x_0$ be a point of the maximum of ${\displaystyle \underset{\left|\xi \right|=1}{max}\left(\right|D^{2}v(x,t)|\xi ,\xi )}$, i.e., $(x_0{e}_1,t)$ is a point of the maximum value of the quadratic form of the absolute value of the operator. This implies, in particular, that $x_0$ is also a maximum point of $v_{11}$.

We will analyze a bound in $(x_0,t_0)$. Roughly speaking, if we complete the orthonormal basis of eigenvectors $(e_1,\dots ,e_N,\dots )$, we know that $v_{1j}=0$, $\forall j⩾0$. The next step is to write the equation solved by $v_{11}$. From (61), we have

$${\left(v_{11}\right)}_t-\Delta \left(v_{11}\right)+F^{\prime }\left(BDv\right)·BD{v}_{11}+D{v}_{1}B{F}^{″}BD{v}_1={\partial }_{11}\left(f\left(Bx\right)\right),$$

(68)

where ${\partial }_{11}\left(f\left(Bx\right)\right)$ is bounded. At each point t, we take a maximum point $x_0$ and apply the maximum principle. At this point, the derivative of the maximum is the derivative at the point of the maximum and therefore we recover that

$${\displaystyle \frac{d}{dt}}\parallel D^2{v\parallel }_{\infty }⩽C_0+|F^{\prime }\left(BDv\right)·BD{v}_{11}+D{v}_{1}B{F}^{″}BD{v}_1|,$$

(69)

since ${\left(v_{11}\right)}_t\left(x_0\right)<0$ and $\Delta \left(v_{11}\right)\left(x_0\right)>0$. The first derivative vanishes; the left-hand side of (69) is regular since we have assumed that ${\partial }_{11}\left(f\left(Bx\right)\right)$ was bounded and therefore we have only the term $D{v}_{1}B{F}^{″}BD{v}_1$ to estimate. This gives

$${\displaystyle \frac{d}{dt}}\parallel D^2{v\parallel }_{\infty }⩽C_0+{\parallel D{v}_{1}B{F}^{″}BD{v}_1\parallel }_{\infty }.$$

(70)

Once we have the above estimate, we direct our attention to the second term on the right-hand side of (70). Note that $D{v}_1$ are vectors and $|D{v}_{1}B{F}^{″}BD{v}_1|$ is a scalar so that the second term in (70) is the inner product of the form $⟨F^{″}BD{v}_1,BB{v}_1⟩$. Hence, we obtain

$${\displaystyle \frac{d}{dt}}\parallel D^2{v\parallel }_{\infty }⩽C_0+\underset{x}{sup}\left\{\right|⟨F^{″}BD{v}_1,BB{v}_1⟩\left|\right\}.$$

(71)

The new requirement of (71) is to look at the matrix $b_{ik}{V}_{kl}$ and its first coordinate $b_{i1}{V}_{11}$ after having diagonalized the matrix $D^{2}u$. We can easily check that

$$v\in \mathrm{BUC}(H\times [0,T]),v\in Lip,|B{D}^{2}vB|⩽C_0,$$

(72)

implies

$$|b_{i1}{V}_{11}{b}_{i1}|⩽C_0.$$

(73)

Accordingly, the estimation (66) leads to (73) in the direction $(1,1)$. In other words, if one looks at the vector $B{e}_1$ multiplied by the weight $\sqrt{V_{11}}$, one has

$$|\sqrt{V_{11}}B{e}_1|⩽C_0.$$

(74)

In particular, the bound established above yields

$$\underset{x}{sup}\left\{\right|⟨F^{″}BD{v}_1,BB{v}_1⟩\left|\right\}⩽C_0{\left|B{e}_1\right|}^2{v}_{11}^2,$$

(75)

and, consequently, the left-hand side of (75) is dominated by

$$\underset{x}{sup}\left\{\right|⟨F^{″}BD{v}_1,BB{v}_1⟩\left|\right\}⩽C_1|v_{11}|=C_1{\parallel D^{2}v\parallel }_{\infty }.$$

(76)

By (76) and (71), one can conclude that

$${\displaystyle \frac{d}{dt}}\parallel D^2{v\parallel }_{\infty }⩽C_0+C_1{\parallel D^{2}v\parallel }_{\infty }$$

(77)

and thus the claim follows. □

Summarizing the arguments of this subsection, we have proven the following result: an inequality that, in general, would be quadratic would eventually lead to explosions, where we cannot eliminate the explosions, thanks to the regularization of the heat equation, and this is enough to transform it into a Gronwall inequality and obtain a real a priori estimate.

5. Some Consequences and Further Results

This section deals with the existence of the viscosity subsolution of the Hamilton–Jacobi–Bellman-type equation (5). We consider a subsequence of increasing subspaces $H_N$ of dimension N and we consider the following problem:

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{\partial u_N}{\partial t}-\Delta u_N+F\left(D{u}_N\right)=f\left(x\right),(x,t)\in H_N\times (0,\infty )}\hfill \\ \hfill & {\displaystyle u_N(x,0)=u_0\left(x\right),}\hfill \end{array}\right.$$

(78)

where $f:H\to \mathbb{R}$ is a known regular function and D is the Fréchet derivative operator in H. Recall that we identify a point of $H_N$ with a point of ${\mathbb{R}}^N$ that shares the same non-zero N-coordinates. The space H is restricted to $H_N$ supplemented by 0, with $F(\nabla u_N)\equiv F(\nabla u_N,0,\dots ,0)$. We do not devise any growth hypothesis on F, but since F is Lipschitz, we do not see the behavior of F at infinity.

Note that all of the a priori formal estimates obtained previously are still true in H and are uniform. As a consequence, we deduce that, possibly at the cost of extracting a subsequence still labelled $u_{N^{\prime }}$ such that

$$u_{N^{\prime }}\underset{N^{\prime }\to \infty }{\overset{ }{\to }}u,$$

we have the B-modulus of continuity on $[0,T]$, the Lipschitz condition, and it satisfies the properties (46), (47), (48) and (49). The only property to prove is the passage to the limit. It is easy to pass to the limit in $F\left(D{u}_N\right)$ since we have compactness with $Du$; the difficult term is the Laplacian. To this end, we will use viscosity solution approaches.

Theorem 2.

There exists a solution to the Cauchy problem (78) satisfying the estimations (46), (47), (48) and (49) and such that the sequence $u_N$ converges uniformly on all compact subsets of H to u.

We recall the classical definition of the subsolution or supersolution (also called the lower and upper solutions).

A bounded, uniformly continuous function $u:(0,T)\times H\to \mathbb{R}$ is called a viscosity subsolution of (78) if, for all $C^2$ functions $\phi :(0,T)\times H\to \mathbb{R}$ such that $u-\phi$ has a local maximum at $(t_0,x_0)$, then

$${\partial }_t\phi -\Delta \phi +F\left(D\phi \right)⩽f.$$

(79)

A bounded, uniformly continuous function $u:(0,T)\times H\to \mathbb{R}$ is called a viscosity supersolution of (78) if, for all $C^2$ functions $\phi :(0,T)\times H\to \mathbb{R}$ such that $u-\phi$ has a local minimum at $(x_0,t_0)$, then

$${\partial }_t\phi -\Delta \phi +F\left(D\phi \right)⩾f.$$

(80)

A viscosity solution of (78) is a function that is both a viscosity subsolution and a supersolution [36].

It remains to clarify the class of the function $\phi$. It is also worth noting that the maxima and minima in the above definition can be assumed to be strict and global and, for bounded sub- and supersolutions, $\phi ,{\phi }_t,D\phi ,D^2\phi$ may be assumed to be bounded and uniformly continuous on $(0,T)\times H$ or belong to $C_b^2$. The additional requirement is

$$|B^{-1}{D}^2\phi B^{-1}{|}_2\mathrm{i}\mathrm{s} \mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d} \mathrm{o}\mathrm{n} \mathrm{H},$$

(81)

which ensures that the Laplacian is a continuous function and that $\Delta \phi \in \mathrm{BUC}\left(H\right)$. It is worth noting that the relation (81) is equivalent to the following one:

$$|D^2\phi |⩽C_{0}A.$$

(82)

The property (82) is sufficient for our test function and gives the sense to the Laplacian. Theorem 2 establishes the existence of viscosity solutions of the problem (78).

It is worth stressing that the results obtained in the present paper can be applied to specific partial differential equations proposed in the literature and, in particular, to optimal control problems; see, among others, the paper [37] and the references therein.