Hamilton–Jacobi–Bellman Equations

Abstract

In this chapter we present recent developments in the theory of Hamilton–Jacobi–Bellman (HJB) equations as well as applications. The intention of this chapter is to exhibit novel methods and techniques introduced few years ago in order to solve long-standing questions in nonlinear optimal control theory of Ordinary Differential Equations (ODEs).

Appendix: Semi-Discrete in Time Approximation Revisited

In this section we review the semi-discrete in time approximation studied in [37], summarized in Sect. 2.5.3, and we improve some of the results therein providing also a complete semi-discrete analysis of analogous results in [41, Sect. 4]. First let us recall that for $w: \mathbb{R}^{n} \rightarrow \mathbb{R}$ the super-differential D ⁺ w(x) at $x \in \mathbb{R}^{n}$ is defined as

$$\displaystyle{ D^{+}w(x):= \left \{p \in \mathbb{R}^{n}\;\ \limsup _{ y\rightarrow x}\frac{w(\,y) - w(x) -\langle \, p,y - x\rangle } {\vert y - x\vert } \leq 0\right \}. }$$

(2.243)

We collect in the following Lemmas some useful properties of semiconcave functions, i.e. functions that satisfy (2.225) (see [39] for a very complete account of this subject).

Lemma 1

For a function $w: \mathbb{R}^{n} \rightarrow \mathbb{R}$ , the following assertions are equivalent:

• (i) The function w is semiconcave, with constant c.

• (ii) For all x, $y \in \mathbb{R}^{n}$ and p ∈ D ⁺ w(x), q ∈ D ⁺ w( y)

  $$\displaystyle{ \langle q - p,y - x\rangle \leq c\vert x - y\vert ^{2}. }$$

  (2.244)

• (iii) Setting I _n for the identity matrix, we have that Δw ≤ cI _n in the sense of distributions.

Lemma 2

Let $w: \mathbb{R}^{n} \rightarrow \mathbb{R}$ be semiconcave. Then:

• (i) w is locally Lipschitz.

• (ii) If w _n is a sequence of uniformly semiconcave functions (i.e. they share the same semiconcavity constant) converging pointwise to w, then the convergence is locally uniform and ∇w _n (⋅ ) → ∇w(⋅ ) a.e. in $\mathbb{R}^{n}$.

2.1.1 Properties of the Semi-Discretization of the HJB Equation

Recall that given h > 0 and $N \in \mathbb{N}$ such that Nh = T, we set t _k : = kh for $k = 0,\mathop{\ldots },N$. Let us define the following spaces:

$$\displaystyle{ \begin{array}{l} \mathcal{K}_{N}:= \left \{\mu = (\mu _{\ell})_{\ell=0}^{N}:\, \quad \mbox{ such that }\mu _{\ell} \in \mathcal{ P}_{1}(\mathbb{R}^{n})\quad \text{for all }\ell = 0,\ldots,N\right \}, \\ \mathcal{A}_{k}:= \left \{\alpha = (\alpha _{\ell})_{\ell=k}^{N-1}:\, \quad \mbox{ such that }\alpha _{\ell} \in \mathbb{R}^{n}\right \}\quad \text{for}\quad k = 0,\ldots,N - 1. \end{array} }$$

For $\mu \in \mathcal{K}_{N}$ and $k = 1,\mathop{\ldots },N$, we consider the following semi-discrete approximation of v[μ] in (2.212)

Classical arguments imply that $(\mathcal{CP})_{h}^{x,k}[\mu ]$ admits at least a solution for all (x, k). We denote by $\mathcal{A}_{k}[\mu ](x) \subseteq \mathcal{A}_{k}$ the set of optimal solutions of $(\mathcal{CP})_{h}^{x,k}[\mu ]$, i.e. the set of discrete optimal controls. Note that v _k [μ](x) can be equivalently defined with the discretized DPP (2.214). Recall also the extension v _h [μ], defined in $\mathbb{R}^{n} \times [0,T]$, of v _k [μ](x) considered in (2.215). We have the following properties for v _h [μ]:

Proposition 7

For all h > 0, we have:

• (i) For any t ∈ [0, T], the function v _h [μ](⋅ , t) is Lipschitz continuous, with a Lipschitz constant c > 0 independent of (μ, h, k).

• (ii) For all t ∈ [0, T] the function v _h [μ](⋅ , t) is semiconcave uniformly in (h, μ, t).

• (iii) There exists a constant c > 0 (independent of (μ, h, x, k)) such that

  $$\displaystyle{ \max _{\ell=k,\mathop{\ldots },N-1}\vert \alpha _{\ell}\vert \leq c\quad \mathit{\mbox{ for all }}\ \alpha \in \mathcal{A}_{k}[\mu ](x). }$$

• (iv) For all $x \in \mathbb{R}^{n}$ , $k = 0,\mathop{\ldots },N - 1$ and $\alpha \in \mathcal{A}_{k}[\mu ](x)$ , we have

  $$\displaystyle{ \alpha _{\ell} + h\nabla F\left (X_{\ell}^{x,k}[\alpha ],\mu _{\ell}\right ) \in D^{+}v_{ h}[\mu ]\left (X_{\ell}^{x,k}[\alpha ],t_{\ell}\right )\quad \mathit{\mbox{ for }}\;\ell = k,\mathop{\ldots },N - 1. }$$

• (v) We have that v _h [μ](⋅ , t) is differentiable at x iff for k = [t∕h] there exists $\alpha \in \mathcal{A}_{k}[\mu ](x)$ such that $\mathcal{A}_{k}[\mu ](x) =\{\alpha \}$ . In that case, the following holds:

  $$\displaystyle{ \nabla v_{h}[\mu ](x,t) =\alpha _{k} + h\nabla F(x,\mu _{k}). }$$

• (vi) Given (x, t) and $\alpha \in \mathcal{A}_{k}[\mu ](x)$ , with k = [t∕h], we have that for all s ∈ [t _k+1, T], the function v _h [μ](⋅ , s) is differentiable at X _ℓ ^x, k[α], with ℓ = [s∕h].

Proof

We only prove (iv) since the other statements are proved in [37]. For notational convenience, we omit the μ argument and we prove the result for ℓ = k, since for $\ell= k + 1,\mathop{\ldots },N$ the assertion follows from (v)–(vi). Let $x,y \in \mathbb{R}^{n}$ and τ ≥ 0. Since $\alpha \in \mathcal{A}_{k}[\mu ](x)$, we have

$$\displaystyle{ v_{k}(x +\tau y) \leq \sum _{\ell=k}^{N-1}\left [\frac{1} {2}\vert \alpha _{\ell}\vert ^{2} + F\left (X_{\ell}^{x+\tau y,k}[\alpha ],\mu _{\ell}\right )\right ]h + G\left (X_{ N}^{x+\tau y,k}[\alpha ],\mu _{ N}\right ), }$$

with equality for τ = 0. Therefore,

$$\displaystyle{ \begin{array}{rcl} v_{k}(x +\tau y) - v_{k}(x)& \leq &h\sum _{\ell=k}^{N-1}\left [F\left (X_{\ell}^{x+\tau y,k}[\alpha ],\mu _{\ell}\right ) - F\left (X_{\ell}^{x,k}[\alpha ],\mu _{\ell}\right )\right ] \\ \ & \ & + G\left (X_{N}^{x+\tau y,k}[\alpha ],\mu _{N}\right ) - G\left (X_{\ell}^{x,k}[\alpha ],\mu _{N}\right ). \end{array} }$$

(2.245)

On the other hand, the optimality condition for α yields

$$\displaystyle{ \alpha _{k} = h\sum _{\ell=k+1}^{N-1}\nabla F\left (X_{\ell}^{x,k}[\alpha ],\mu _{\ell}\right ) + \nabla G\left (X_{\ell}^{x,k}[\alpha ],\mu _{ N}\right ). }$$

Combining with (2.245) and taking the limit as τ → 0, gives

$$\displaystyle{ \limsup _{\tau \rightarrow 0}\frac{v_{k}(x +\tau y) - v_{k}(x)} {\tau } -\langle \alpha _{k} + h\nabla F(x,\mu _{k}),y\rangle \leq 0, }$$

which, by Cannarsa and Sinestrari [39, Proposition 3.15 and Theorem 3.2.1], implies the result.

Given (x, k) and $\alpha \in \mathcal{A}_{k}[\mu ](x)$ we set

$$\displaystyle{ \alpha _{k}[\mu ](x):=\alpha _{k}. }$$

(2.246)

Proposition 7(iv) implies that

$$\displaystyle{ \alpha _{k}[\mu ](x) \in D^{+}v_{ h}[\mu ](x,t_{k}) - h\nabla F(x,u_{k}). }$$

(2.247)

A straightforward computation shows that α _k [μ](x) solves, for each (x, k), the problem defined in the r.h.s. of (2.214). Moreover, by Proposition 7(v)–(vi), the following relation holds true

$$\displaystyle{ \alpha _{\ell} =\alpha _{\ell}[\mu ]\left (X_{\ell}^{x,k}[\alpha ]\right )\quad \mbox{ for all }\;\;\;\ell = k,\mathop{\ldots },N - 1. }$$

(2.248)

2.1.2 Semi-Discretization of the Continuity Equation

Let $\alpha ^{x,k}[\mu ] \in \mathcal{A}_{k}$ be a measurable selection of the multifunction $(x,k) \rightarrow \mathcal{A}_{k}[\mu ](x)$. Given this measurable selection, we set α _k [μ](x) = α _k ^x, k[μ], as in (2.246). By (2.247)– (2.248), there exists a measurable function $(x,k) \rightarrow p_{k}[\mu ](x) \in \mathbb{R}^{n}$ such that p _k [μ](x) ∈ D ⁺ v _k [μ](x) and for all time iterations $\ell= k,\mathop{\ldots },N$ we have

$$\displaystyle{ \alpha _{\ell}[\mu ]\left (X_{\ell}^{x,k}[\alpha ^{x,k}[\mu ]]\right ) = p_{\ell}[\mu ]\left (X_{\ell}^{x,k}[\alpha ^{x,k}[\mu ]]\right ) - h\nabla F\left (X_{\ell}^{x,k}[\alpha ^{x,k}[\mu ]],\mu _{\ell}\right ). }$$

(2.249)

Moreover, Proposition 7(v)–(vi) implies that for $\ell= k + 1,\mathop{\ldots },N$

$$\displaystyle{ p_{\ell}[\mu ]\left (X_{\ell}^{x,k}[\alpha _{\ell}^{x,k}[\mu ]\right ) = \nabla v_{\ell}[\mu ]\left (X_{\ell}^{x,k}[\alpha _{\ell}^{x,k}[\mu ]]\right )\quad \mbox{ for all }x \in \mathbb{R}^{n} }$$

(2.250)

and

$$\displaystyle{ p_{k}[\mu ]\left (x\right ) = \nabla v_{k}[\mu ]\left (x\right )\quad \mbox{ for a.a. }x \in \mathbb{R}^{n}. }$$

(2.251)

Given (x, k ₁), the discrete flow $\varPhi _{k_{1},\cdot }[\mu ](x) \in \mathbb{R}^{(N-k)\times n}$ is defined as

$$\displaystyle{ \varPhi _{k_{1},k_{2}}[\mu ](x):= x - h\sum _{\ell=k_{1}}^{k_{2}-1}\alpha _{ \ell}^{x,k_{1} }[\mu ]\quad \mbox{ for all }k_{2} \geq k_{1}. }$$

(2.252)

Equivalently, by (2.249), for all k ₁ ≤ k ₂ ≤ k ₃,

$$\displaystyle{ \begin{array}{rcl} \varPhi _{k_{1},k_{3}}[\mu ](x)&:=&x - h\sum _{\ell=k_{1}}^{k_{3}-1}\alpha _{\ell}[\mu ]\left (X_{\ell}^{x,k_{1}}[\alpha ^{x,k_{1}}[\mu ]]\right ), \\ \ & = &\varPhi _{k_{1},k_{2}}[\mu ](x) - h\sum _{\ell=k_{2}}^{k_{3}-1}\alpha _{\ell}[\mu ]\left (X_{\ell}^{x,k_{1}}[\alpha ^{x,k_{1}}[\mu ]]\right ).\end{array} }$$

(2.253)

In particular, for all k ₁ ≤ k ₂,

$$\displaystyle{ \varPhi _{k_{1},k_{2}+1}[\mu ](x) =\varPhi _{k_{1},k_{2}}[\mu ](x) - h\alpha _{k_{2}}[\mu ]\left (\varPhi _{k_{1},k_{2}}[\mu ](x)\right ). }$$

(2.254)

The following result is an important improvement of [37, Lemma 3.6].

Proposition 8

There exists a constant c > 0 (independent of μ and small enough h) such that for all k = 1, …, N and $x,y \in \mathbb{R}^{n}$ we have

$$\displaystyle{ \vert \varPhi _{0,k}[\mu ](x) -\varPhi _{0,k}[\mu ](\,y)\vert \geq c\vert x - y\vert. }$$

(2.255)

Thus, Φ _0,k [μ](⋅ ) is invertible in $\varPhi _{0,k}[\mu ](\mathbb{R}^{n})$ and the inverse Υ _0,k [μ](⋅ ) is 1∕c-Lipschitz.

Proof

For notational convenience, let us set Φ _k = Φ _0,k [μ](x) and Ψ _k = Φ _0,k [μ]( y). Expression (2.254) implies that

$$\displaystyle{ \left \vert \varPhi _{k+1} -\varPsi _{k+1}\right \vert ^{2} \geq \left \vert \varPhi _{ k} -\varPsi _{k}\right \vert ^{2} - 2h\left [\alpha _{ k}[\mu ](\varPhi _{k}) -\alpha _{k}[\mu ](\varPsi _{k})\right ] \cdot (\varPhi _{k} -\varPsi _{k}). }$$

(2.256)

By (2.249) we have (omitting the dependence on μ)

$$\displaystyle{ \alpha _{k}(\varPhi _{k}) -\alpha _{k}(\varPsi _{k}) = p_{k}(\varPhi _{k}) - p_{k}(\varPsi _{k}) - h\left [\nabla F(\varPhi _{k}) -\nabla F(\varPsi _{k})\right ]. }$$

Using the semiconcavity of v _k [μ](⋅ ) and the fact that F has bounded second order derivatives w.r.t. x, Lemma 1(iii) gives

$$\displaystyle{ \left [\alpha _{k}(\varPhi _{k}) -\alpha _{k}(\varPsi _{k})\right ] \cdot (\varPhi _{k} -\varPsi _{k}) \leq c(1 + h)\left \vert \varPhi _{k} -\varPsi _{k}\right \vert ^{2}, }$$

(2.257)

for some c > 0. By (2.256) and (2.257), there is c ^′ > 0 (independent of h small enough) such that

$$\displaystyle{ \left \vert \varPhi _{k+1} -\varPsi _{k+1}\right \vert ^{2} \geq (1 - hc^{{\prime}})\left \vert \varPhi _{ k} -\varPsi _{k}\right \vert ^{2}. }$$

Therefore, for every k = 1, …, N, we get

$$\displaystyle{ \left \vert \varPhi _{k+1} -\varPsi _{k+1}\right \vert ^{2} \geq (1 - hc^{{\prime}})^{k}\left \vert x - y\right \vert ^{2} \geq (1 - hc^{{\prime}})^{[T/h]}\left \vert x - y\right \vert ^{2}. }$$

and the result follows from the convergence of (1 − hc ^′)^[T∕h] to exp(−c ^′ T) as h ↓ 0.

As we already explained in Sect. 2.5.3, a natural semi-discretization of the solution m[μ] of (2.226) is obtained as the push-forward of m ₀ under the discrete flow Φ _0,k [μ](⋅ ). For every $k = 0,\mathop{\ldots },N$ set

$$\displaystyle{ m_{k}[\mu ]:=\varPhi _{0,k}[\mu ](\cdot )\sharp m_{0}. }$$

(2.258)

By (2.253) we have

$$\displaystyle{ m_{k}[\mu ] =\varPhi _{\ell,k}[\mu ](\cdot )\sharp m_{\ell}[\mu ]\quad \mbox{ for all }\ell = 1,\mathop{\ldots },k, }$$

(2.259)

In particular, for all $\phi \in C_{b}(\mathbb{R}^{n})$ we have

$$\displaystyle{ \int _{\mathbb{R}^{n}}\phi (x)\mathrm{d}m_{k+1}[\mu ](x) =\int _{\mathbb{R}^{n}}\phi \left (x - h\alpha _{k}[\mu ](x)\right )\mathrm{d}m_{k}[\mu ](x), }$$

(2.260)

which applied with ϕ ≡ 1 gives $m_{k}[\mu ](\mathbb{R}^{n}) = 1$ for $k = 0,\mathop{\ldots },N$.

We have the following Lemma, which improves [37, Lemma 3.7] since we now prove, using Proposition 8, uniform bounds for the density of m _k [μ]. Recall the distance d ₁(μ, ν) between to probability measures with finite first order moments is defined in (2.209).

Lemma 3

There exists c > 0 (independent of (μ, h)) such that:

• (i) For all k ₁, k ₂ ∈ {1, …, N}, we have that

  $$\displaystyle{ d_{1}(m_{k_{1}}[\mu ],m_{k_{2}}[\mu ]) \leq ch\vert k_{1} - k_{2}\vert = c\vert t_{k_{1}} - t_{k_{2}}\vert. }$$

  (2.261)

• (ii) For all k = 1, …, N, m _k [μ] is absolutely continuous (with density still denoted by m _k [μ]), has a support in B(0, c) and ∥m _k [μ]∥ _∞ ≤ c.

Proof

By Proposition 7(iii) we have

$$\displaystyle{ \vert \varPhi _{0,k_{1}}[\mu ](x) -\varPhi _{0,k_{2}}[\mu ](x)\vert \leq ch\vert k_{1} - k_{2}\vert = c\vert t_{k_{1}} - t_{k_{2}}\vert. }$$

(2.262)

By definition of m _k [μ](⋅ ), we have that for any 1-Lipschitz function $\phi: \mathbb{R}^{n} \rightarrow \mathbb{R}$

$$\displaystyle{ \begin{array}{ll} \int _{\mathbb{R}^{n}}\phi (x)\,d\left [m_{k_{1}}[\mu ] - m_{k_{2}}[\mu ]\right ](x)& \leq \int _{\mathbb{R}^{n}}\vert \varPhi _{0,k_{1}}[\mu ](x) -\varPhi _{0,k_{2}}[\mu ](x)\vert \mathrm{d}m_{0}(x) \\ \ & \leq ch\vert k_{1} - k_{2}\vert = c\vert t_{k_{1}} - t_{k_{2}}\vert.\end{array} }$$

On the other hand, since by (H1) we have $\mathop{\mathrm{supp}}(m_{0}) \subset B(0,c_{1})$, Proposition 7(iii) implies that $\mathop{\mathrm{supp}}(m_{k}[\mu ])$ is contained in B(0, c ^′) for some c ^′ > 0. Moreover, for any Borel set A and k = 1, …, N, Proposition 8 and the fact that ∥m ₀∥ _∞ ≤ c imply the existence of c ^′′ > 0 such that

$$\displaystyle{ m_{k}[\mu ](A) = m_{0}(\varUpsilon _{0,k}[\mu ](A)) \leq \| m_{0}\|_{\infty }\vert \varUpsilon _{0,k}(A)\vert \leq c^{{\prime\prime}}\vert A\vert, }$$

where | A | denotes the Lebesgue measure of the set A. Thus, m _k [μ] is absolutely continuous and its density, still denoted by m _k [μ], satisfies ∥m _k [μ]∥ _∞ ≤ c ^′′. The result easily follows.

Recall that m _k [μ] is extended to an element m _h [μ](⋅ ) of $C([0,T];\mathcal{P}_{1}(\mathbb{R}^{n}))$ as in (2.221). The following result is a clear consequence of Lemma 3 and (2.221).

Proposition 9

There exists a constant c > 0 (independent of μ and small enough h ) such that:

• (i) For all t ₁, t ₂ ∈ [0, T], we have that

  $$\displaystyle{ d_{1}(m_{h}[\mu ](t_{1}),m_{h}[\mu ](t_{2})) \leq c\vert t_{1} - t_{2}\vert. }$$

  (2.263)

• (ii) For all t ∈ [0, T], m _h [μ](t) is absolutely continuous (with density denoted by m _h [μ](⋅ , t)), has a support in B(0, c) and ∥m _h [μ](⋅ , t)∥ _∞ ≤ c.

2.1.3 The Semi-Discrete Scheme for the First Order MFG Problem

Recall that the semidiscretization of the MFG problem (2.210) can be written as

$$\displaystyle{ \mbox{ Find }\mu \in C([0,T];\mathcal{P}_{1}(\mathbb{R}^{n}))\mbox{ such that }\;\mu = m_{ h}[\mu ]. }$$

(2.264)

The following result is proved in [37].

Theorem 12

Under our assumptions we have that  (2.264) admits at least one solution $m_{h} \in \mathcal{K}_{N}$ . Moreover, if the following monotonicity conditions hold true

$$\displaystyle{ \begin{array}{l} \int _{\mathbb{R}^{n}}\left [F(x,m_{1}) - F(x,m_{2})\right ]\mathrm{d}[m_{1} - m_{2}](x) > 0\quad \mathit{\mbox{ for all }}\;m_{1}\neq m_{2} \in \mathcal{ P}_{1} \\ \int _{\mathbb{R}^{n}}\left [G(x,m_{1}) - G(x,m_{2})\right ]\mathrm{d}[m_{1} - m_{2}](x) > 0\quad \mathit{\mbox{ for all }}\,m_{1},m_{2} \in \mathcal{ P}_{1},\;m_{1}\neq m_{2}.\end{array} }$$

(2.265)

then the solution is unique.

Remark 5

The monotonicity assumption (2.265) has been proposed in [94] and is also a sufficient condition to guarantee the uniqueness of a solution of (2.210).

Now, let us prove the convergence of solutions m _h of (2.264).

Theorem 13

Under our assumptions, as h ↓ 0 every limit point of m _h in $C([0,T];\mathcal{P}_{1})$ (there exists at least one) solves  (2.264). In particular, if  (2.265) holds true, we have that m _h → m (the unique solution of  (2.264)) in $C([0,T];\mathcal{P}_{1})$ and in $L^{\infty }\left (\mathbb{R}^{n} \times [0,T]\right )$ -weak-∗.

Proof

Proposition 9 and Ascoli Theorem imply that m _h has at least one limit point $\bar{m}$ in $C([0,T];\mathcal{P}_{1})$ as h ↓ 0. Now, setting v _h = v _h [m _h ] classical arguments (see e.g. [56, 57]) imply that $v_{h} \rightarrow v[\bar{m}]$ uniformly on compact sets of $\mathbb{R}^{n} \times (0,T)$, where $v[\bar{m}]$ is the unique viscosity solution of (2.211) (with $\mu =\bar{ m}$).

In order to conclude the proof we have to show that $\bar{m} = m[\bar{m}] =\varPhi [\bar{m}](0,t,\cdot )\sharp m_{0}$ (recall the notations introduced in Sect. 2.5.3). By Cardaliaguet [41, Sect. 4] there exists a set $A \subseteq \mathbb{R}^{n}$, with $\mathcal{L}^{n}(\mathbb{R}^{n}\setminus A) = 0$, such that

$$\displaystyle{ \mathcal{A}(x,0):=\{ \mbox{ the set of optimal controls for }v[\bar{m}](x,0)\}, }$$

is a singleton {α(x, 0)} for all x ∈ A. Proposition 7(iii) implies that the optimal controls $\bar{\alpha }_{h}(x,\cdot )$, associated with v _h (x, 0), are bounded in $L^{\infty }([0,T]; \mathbb{R}^{n})$ (in particular they are bounded in $L^{2}([0,T]; \mathbb{R}^{n})$). Thus, up to subsequence, α _h (x, ⋅ ) converges weakly in $L^{2}([0,T); \mathbb{R}^{n})$ to some $\bar{\alpha }(x,\cdot )$ and thus the flow Φ _h (x, ⋅ ): = x − ∫ ₀ ^⋅ α _h (x, s)ds converge uniformly to some $\bar{\varPhi }(x,\cdot ):= x -\int _{0}^{\cdot }\bar{\alpha }(x,s)\mathrm{d}s$. Now, since $v_{h}(x,0) \rightarrow v[\bar{m}](x,0)$, by the uniqueness of the optimal control for x ∈ A, we obtain that $\bar{\alpha }=\alpha (x,0)$. Using that m ₀ has compact support, by the dominated convergence theorem, the convergence for x ∈ A of Φ _h (x, ⋅ ) to $\bar{\varPhi }(x,\cdot )$, which is the optimal trajectory for $v[\bar{m}](x,0)$, the optimal yields that for every 1-Lipschitz $\varphi: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and t ∈ [0, T],

$$\displaystyle{ \int _{\mathbb{R}^{n}}\varphi (x)\mathrm{d}[m_{h}(t) - m[\bar{m}](t)](x) \leq \int _{\mathbb{R}^{n}}\vert \varPhi _{h}(x,t) -\bar{\varPhi } (x,t)\vert m_{0}(x)\mathrm{d}x \rightarrow 0\quad \mbox{ as }h \downarrow 0. }$$

Therefore $\bar{m} = m[\bar{m}]$, which ends the proof. Finally, by Proposition 9(ii) we have that $m_{h} \rightarrow \bar{ m}$ in $L^{\infty }\left (\mathbb{R}^{n} \times [0,T]\right )$-weak-∗. The result follows.