A fully backward representation of semilinear PDEs applied to the control of thermostatic loads in power systems

Lemma A.1.

We recall that b ~ was defined in (3.10). Let W be an ( F t ) t ∈ [ 0 , T ] -Brownian motion and let X be a solution of

where X 0 is a Gaussian random vector independent of W. Set L t := σ ⁢ ( t ) - 1 ⁢ b ~ ⁢ ( t , X t ) for all t ∈ [ 0 , T ] . Then the Doléans exponential

is an ( F t ) t ∈ [ 0 , T ] -martingale.

Proof.

By following [25, Corollary 5.14], it is sufficient to find a constant time step subdivision ( t n ) n ∈ ℕ of [ 0 , T ] such that, for all n ∈ ℕ ,

By combining Jensen’s inequality and Fubini’s theorem, this is fulfilled in particular if for all n ∈ ℕ ,

where δ := t n + 1 - t n . Let s ∈ [ 0 , T ] . Then

since a , c are bounded and σ - 1 is also bounded being continuous on [ 0 , T ] . Furthermore, by Lemma 3.8 (i) and (3.14), X is a Gaussian process with mean function m X (resp. covariance function Q X ) solving the first line of equation (3.1) (resp. (3.2)) with initial condition 𝔼 ⁢ ( X 0 ) (resp. Cov ⁢ ( X 0 ) ).

By taking into account the fact that m X is bounded (since it is continuous), it suffices to find a subdivision such that

where Z ∼ 𝒩 ⁢ ( 0 , I d ) and K := 4 ⁢ ∥ σ - 1 ∥ ∞ 2 ⁢ ∥ a ∥ ∞ 2 ⁢ ∥ Q X ∥ ∞ > 0 . This is the case in particular if K ⁢ δ < 1 , which ends the proof. ∎

Lemma A.2.

Suppose the validity of Assumption 4.1. Suppose in addition that the functions g and

are locally Lipschitz with polynomial growth gradient (uniformly in t and α). Then, for each ( t , α ) ∈ [ 0 , T ] × A 0 ,

is locally Lipschitz, uniformly in t and α.

Proof.

We give here a proof of the local Lipschitz property for the term involving the function g since the other term can be treated in the same way.

Let ( t , α ) ∈ [ 0 , T ] × 𝒜 0 and let x , y be in a compact set of ℝ d . Let K be the Lipschitz constant of b. Using in particular the Cauchy–Schwarz inequality, we get

where we have used the estimate | X T t , x , α - X T t , y , α | ≤ e K ⁢ T ⁢ | x - y | , following from the identity

together with Gronwall’s lemma. In view of (A.1), the point is proved if

is bounded uniformly in t , x , y , α . This follows from polynomial growth of ∇ x ⁡ g , classical moment estimates for sup s ∈ [ t , T ] ⁡ | X s t , z , α | , z ∈ ℝ d (see for example [26, Corollary 2.5.12]), and the fact x , y lie in a compact set. ∎

Lemma A.3.

Let Λ be an arbitrary set and let O be an open subset of R d . Let x ∈ R d . Let F : O × Λ ↦ R such that, for all λ ∈ Λ , F ⁢ ( ⋅ , λ ) and V : x ↦ sup λ ∈ Λ ⁡ F ⁢ ( x , λ ) are differentiable at the point x. Suppose also that Λ * ⁢ ( x ) = { λ ∈ Λ : V ⁢ ( x ) = F ⁢ ( x , λ ) } is not empty. Then

for every λ x * ∈ Λ * ⁢ ( x ) .

Proof.

Let x be as in the proposition statement and let h ∈ ℝ d . Let λ x * ∈ Λ * ⁢ ( x ) . Then, using in particular the differentiability of F ⁢ ( ⋅ , λ x * ) at the point x, we get

By the differentiability of V at the point x, (A.2) implies

Setting h to - h in (A.2) and proceeding as before, we obtain

Combining (A.3) and (A.4), we get

which forces ∇ x ⁡ V ⁢ ( x ) = ∇ x ⁡ F ⁢ ( x , λ x * ) . This ends the proof. ∎