Stability analysis of the Navier–Stokes velocity tracking problem with bang-bang controls

Abstract

This paper focuses on the stability of solutions for a velocity-tracking problem associated with the two-dimensional Navier–Stokes equations. The considered optimal control problem does not possess any regularizer in the cost, and hence bang-bang solutions can be expected. We investigate perturbations that account for uncertainty in the tracking data and the initial condition of the state, and analyze the convergence rate of solutions when the original problem is regularized by the Tikhonov term. The stability analysis relies on the Hölder subregularity of the optimality mapping, which stems from the necessary conditions of the problem.

Appendix

We now collect the stability results of the paper in an abstract framework, as the same principles can be applied to other types of optimization problems. We employ normed spaces since they constitute an adequate setting for our purposes; norms provide positively homogeneous measures for notions of growth and convergence, unlike general abstract metric spaces.

The results of this section focus mainly on necessary and sufficient conditions for stability of the first-order necessary conditions in optimization. For convenience of the reader, this section is intended to be absolutely self-contained and independent of other sections. Throughout the Appendix, unless otherwise stated, $\big (U,\Vert \cdot \Vert _{U}\big )$ is a normed space and $\mathcal U$ a convex subset of U. We also consider a real-valued functional $\mathcal J:\mathcal U\rightarrow \mathbb R$. We will focus on stability properties associated to the minimization problem

$$\begin{aligned} \min _{u\in \mathcal U}\mathcal J(u). \end{aligned}$$

(35)

In the context of optimal control, $\mathcal U$ is to be interpreted as the set of controls, and $\mathcal J:\mathcal U\rightarrow \mathbb R$ as the objective functional. We will see that the stability of the system of necessary conditions for problem (35) is closely related to the growth conditions satisfied by $\mathcal J$ at a local minimizer.

1.1 A first-order variant of the Ekeland principle

The first subsection is of technical nature and is devoted to recalling a few results of variational analysis that will be used later on. In particular, we state a first-order variant of the seminal Ekeland variational principle.

We begin recalling the standard notion of (first-order) Gateaux differentiability. We say that $\mathcal J$ is Gateaux differentiable at $\bar{u}\in \mathcal U$ if there exists a linear mapping $\mathcal J'(u)\in U^*$ such that

$$\begin{aligned} \mathcal {J}'(\bar{u})v=\lim _{\varepsilon \rightarrow 0^+}\frac{{\mathcal J}(\bar{u}+\varepsilon v)-\mathcal {J}(\bar{u})}{\varepsilon }\quad \forall v\in \mathcal U-u. \end{aligned}$$

Working with functions defined on convex domains has the advantage of simple tangent and normal cone formulations, which in turn implies that the first-order necessary condition also take a simpler form.

The normal cone to $\mathcal U$ at $\bar{u}$ is defined by

$$\begin{aligned} N_\mathcal {U}(\bar{u}):=\left\{ \rho \in U^*:\text { }\rho (u-\bar{u})\le 0\quad \text {for all}\,u\in \mathcal U\right\} . \end{aligned}$$

(36)

The first-order necessary condition is well-known for Gateaux differentiable functions, and a lot of the work carried out in optimization and variational analysis relies on it; see [22, pp. 11-13]. If $\mathcal {J}$ is Gateaux differentiable at a local minimizer $\bar{u}\in \mathcal U$, then $0\in \mathcal J'(\bar{u})+N_\mathcal {U}(\bar{u})$. Second-order necessary conditions for optimality are also well known; see, e.g., [5, Lemma 3.44] or [35, Theorem 3.45].

We give now a technical lemma based on the celebrated Sion Minimax Theorem.

Lemma A.1

Let $\psi :\mathcal {U}\rightarrow \mathbb R$ be a convex lower semicontinuous function. Let $\hat{u}\in U$ and $\gamma >0$. There exists ${\hat{\rho }}\in U^*$ with $\Vert {\hat{\rho }}\Vert _{U^*}\le \gamma$ such that

$$\begin{aligned} \inf _{u\in M} \Big \{\psi (u) + \gamma \Vert u-\hat{u}\Vert _U\Big \} = \inf _{u\in \mathcal {U}}\left\{ \psi (u)-{\hat{\rho }} (u-\hat{u})\right\} . \end{aligned}$$

Proof

Let $\mathbb B^*$ be the unit ball of $U^*$. Define $f:\mathcal {U}\times \mathbb {B}^*\rightarrow \mathbb R$ by $f(u,\rho ):=\psi (u)+\gamma \rho (u-\hat{u})$. Note that $U^*$ endowed with the weak* topology is a linear topological space, and $\mathbb B^*$ is weak* compact by Banach-Alaoglu Theorem. The function $f(\cdot ,\rho ):\mathcal {U}\rightarrow \mathbb R$ is convex and lower semicontinuous for each $\rho \in U^*$. The function $f(u,\cdot ): \mathbb {B}^*\rightarrow \mathbb R$ is weak* continuous and affine for each $u\in \mathcal {U}$. The hypotheses of Sion Minimax Theorem [37, Corollary 3.3] are then satisfied, and hence

$$\begin{aligned} \inf _{u\in \mathcal {U}}\sup _{\rho \in \mathbb B^*}\left\{ \psi (u)+\gamma \rho (u-\hat{u})\right\} =\sup _{\rho \in \mathbb B^*}\inf _{u\in \mathcal {U}} \{\psi (u)+\gamma \rho (u-\hat{u})\}. \end{aligned}$$

(37)

Let $h:\mathbb B^*\rightarrow \mathbb R$ be given by $h(\rho ):=\inf _{u\in \mathcal {U}}\{\psi (u)+\gamma \rho (u-\hat{u})\}$. Clearly, h is weak* upper semicontinuous as it is the infimum of weak* continuous functions; and since $\mathbb B^*$ is weak* compact, there exists $\rho ^*\in \mathbb B^*$ such that $\sup _{\rho \in \mathbb B^*} h(\rho )=h(\rho ^*)$. This implies

$$\begin{aligned} \sup _{\rho \in \mathbb B^*}\inf _{u\in \mathcal {U}}\{\psi (u)+\gamma \rho (u-\hat{u})\}=\inf _{u\in \mathcal {U}}\{\psi (u)+\gamma \rho ^*(u-\hat{u})\}. \end{aligned}$$

(38)

Finally, by (37) and (38),

$$\begin{aligned} \inf _{u\in \mathcal {U}} \Big \{\psi (u) + \gamma \Vert u-\hat{u}\Vert _U\Big \}&=\inf _{u\in \mathcal {U}}\sup _{\rho \in \mathbb B^*}\Big \{\psi (u) +\gamma \rho (u-\hat{u})\Big \}\\&=\inf _{u\in \mathcal {U}}\Big \{\psi (u) +\gamma \rho ^*(u-\hat{u})\Big \}. \end{aligned}$$

The results follow defining ${\hat{\rho }}:=-\gamma \rho ^*$.$\square$

We can now prove the following variant of Ekeland principle.

Lemma A.2

Suppose $\big (U,\Vert \cdot \Vert _{U}\big )$ is a Banach space, $\mathcal U$ is a closed convex subset of U, and $\mathcal {J}:\mathcal U\rightarrow \mathbb R$ is a lower semicontinuous Gateaux differentiable function. Let $\bar{u}\in U$ and $r>0$ such that

$$\begin{aligned} \mathcal {J}(\bar{u})\le \mathcal {J}(s)\quad \text {for all}\, s\in \mathcal U\hbox { with }\Vert s-\bar{u}\Vert _{U}\le r. \end{aligned}$$

Let $u\in \mathcal U$ and $\varepsilon >0$ satisfy $\Vert u-\bar{u}\Vert _{U}<r$ and $\mathcal {J}(u)\le \mathcal {J}(\bar{u})+\varepsilon .$ Then for every $\lambda \in \big (0,r-\Vert u-\bar{u}\Vert _U\big )$ there exist $\hat{u}\in \mathcal U$ and ${\hat{\rho }}\in U^*$ such that

• (i) $\Vert u-\hat{u}\Vert _{U}\le \lambda$;

• (ii) $\displaystyle \Vert {\hat{\rho }}\Vert _{U^*}\le \frac{\varepsilon }{\lambda }$;

• (iii) ${\hat{\rho }}\in \mathcal J'(\hat{u})+N_\mathcal {U}(\hat{u})$.

Proof

Let $S:=\{s\in \mathcal U: \Vert s-\bar{u}\Vert _{U}\le r\}$. Since S is a closed subset of U, S is a complete metric space endowed with the metric induced from the norm of U, and $\mathcal {J}|_S$ is lower semicontinuous. We can apply Ekeland Principle [24, Theorem 1.1] to obtain that for every $\lambda \in \big (0,r-\Vert u-\bar{u}\Vert _U\big )$ there exists $\hat{u}\in S$ such that

• (a) $\Vert u-\hat{u}\Vert _{U}\le \displaystyle \lambda$;

• (b) $\mathcal {J}(\hat{u})\le \mathcal {J}(u)$;

• (c) $\hat{{\mathcal {J}}}(\hat{u})\le \hat{{\mathcal {J}}}(s)$ for all $s\in S$, where $\hat{{\mathcal {J}}}:\mathcal U\rightarrow \mathbb R$ is given by

  $$\begin{aligned} \hat{{\mathcal {J}}}(s):={\mathcal J}(s)+\frac{\varepsilon }{\lambda }\Vert s-\hat{u}\Vert _{U}. \end{aligned}$$

Let $r_{\lambda }:=r-\Vert u-\bar{u}\Vert _{U}-\lambda$; clearly $r_{\lambda }>0$. If $s\in \mathcal U$ satisfies $\Vert s-\hat{u}\Vert _{U}\le r_{\lambda }$, then

$$\begin{aligned} \Vert s-\bar{u}\Vert _{U}\le \Vert s-\hat{u}\Vert _{U}+ \Vert \hat{u}-u\Vert _{U}+ \Vert u-\bar{u}\Vert _{U}\le r_{\lambda }+\lambda +\Vert u-\bar{u}\Vert _{U}=r. \end{aligned}$$

Thus, from item (c), $\hat{{\mathcal {J}}}(\hat{u})\le \hat{{\mathcal J}}(s)$ for all $s\in \mathcal U$ with $\Vert s-\hat{u}\Vert _{U}\le r_{\lambda }$. We conclude that $\hat{u}$ is a local minimizer of $\hat{{\mathcal {J}}}$. From this, we get

$$\begin{aligned} 0\le \liminf _{t\rightarrow 0^+}\hspace{0.05cm}\frac{\hat{{\mathcal {J}}}(\hat{u}+t(s-\hat{u}))-\hat{{\mathcal {J}}}(\hat{u})}{t}=\mathcal J'(\hat{u})(s-\hat{u})+\frac{\varepsilon }{\lambda }\Vert s-\hat{u}\Vert _{U}\quad \forall s\in \mathcal U. \end{aligned}$$

This can be rewritten as

$$\begin{aligned} 0=\inf _{s\in \mathcal U}\Big \{\mathcal J'(\hat{u})(s-\hat{u})+\frac{\varepsilon }{\lambda }\Vert s-\hat{u}\Vert _{U}\Big \} \end{aligned}$$

By Lemma A.1, there exists ${\hat{\rho }}\in U^*$ with $\Vert {\hat{\rho }}\Vert _{U^*}\le \varepsilon /\lambda$ such that

$$\begin{aligned} 0&=\inf _{s\in \mathcal U}\Big \{\mathcal J'(\hat{u})(s-\hat{u})+\frac{\varepsilon }{\lambda }\Vert s-\hat{u}\Vert _{U}\Big \}\\&=\inf _{s\in \mathcal U}\Big \{\mathcal J'(\hat{u})(s-\hat{u})-{\hat{\rho }}(s-\hat{u})\Big \}\le \mathcal J'(\hat{u})(v-\hat{u})-{\hat{\rho }}(v-\hat{u}) \end{aligned}$$

for all $v\in \mathcal U$. This implies ${\hat{\rho }}\in \mathcal J'(\hat{u})+N_\mathcal {U}(\hat{u})$. Clearly, $\hat{u}$ and ${\hat{\rho }}$ satisfy items (i)-(iii).$\square$

1.2 Strong Hölder subregularity of the optimality mapping

This subsection is devoted to study the behavior of critical points under the presence of perturbations. We derive necessary and sufficient conditions for stability of the variational inequality describing the first-order necessary condition at critical points. From this point on, we assume that $\mathcal {J}:\mathcal {U}\rightarrow \mathbb {R}$ is Gateaux differentiable, unless we specify otherwise.

Stability of the first-order necessary conditions. In the literature, the stability of the first-order necessary conditions is studied as a property of a set-valued mapping encapsulating the (generalized) equation satisfied by local minimizers. This property is known as strong Hölder (metric) subregularity, see [23, Section 3I]; the property has also appeared in the literature by the name of strong (metric) $\theta$-subregularity, see [16, Section 4].

Let us begin giving a suitable notion for the correspondence between solutions of the perturbed variational inequality and the perturbations. The set-valued mapping $\Phi :\mathcal U \twoheadrightarrow U^*$ given by

$$\begin{aligned} \Phi (u):=\mathcal J'(u)+N_\mathcal {U}(u) \end{aligned}$$

is called the optimality mapping. We now give the definition of stability that we wish to analyze, i.e., the so-called strong (metric) subregularty.

Definition A.1

Let $\bar{u}\in \mathcal U$ satisfy $0\in \Phi (\bar{u})$. We say that the optimality mapping $\Phi :\mathcal U\twoheadrightarrow U^*$ is strongly (Hölder) subregular at $\bar{u}$ (with exponent $\theta \in (0,\infty )$) if there exist positive numbers $\alpha$ and $\kappa$ such that the following property holds. For all $u\in \mathcal U$ and $\rho \in U^*$,

$$\begin{aligned} \Vert u-\bar{u}\Vert _U\le \alpha \quad \text {and}\quad \rho \in \mathcal J' (u)+N_\mathcal {U}(u)\quad \text {imply}\quad \Vert u-\bar{u}\Vert _{U}\le \kappa \Vert \rho \Vert _{U^*}^{\theta }. \end{aligned}$$

(39)

We now proceed to state both sufficient and necessary conditions for this notion of stability.

Sufficient conditions. The proof of the sufficient condition for stability, as shown in the next theorem, follows the arguments presented in [21, Theorem 1] to the letter, where it was previously proven in the context of optimal control.

Theorem A.1

Let $\bar{u}\in \mathcal U$ such that $0\in \Phi (\bar{u})$, and $\mu \in (0,\infty )$. Suppose there exist positive numbers $\delta$ and c such that

$$\begin{aligned} \mathcal J'(u)(u-\bar{u})\ge c\Vert u-\bar{u}\Vert _{U}^{\mu +1}\quad \text {for all}\,u\in \mathcal U\hbox { with }\Vert u-\bar{u}\Vert _{U}\le \delta . \end{aligned}$$

(40)

Then the optimality mapping $\Phi :\mathcal U\twoheadrightarrow U^*$ is strongly Hölder subregular at $\bar{u}$ with exponent $1/\mu$.

Proof

Let $u\in \mathcal U$ and $\rho \in U^*$ be arbitrary satisfying $\Vert u-\bar{u}\Vert _{U}\le \delta$ and $\rho \in \Phi (u)$. Then, as $\rho -\mathcal J'(u)\in N_\mathcal {U}(u)$ and $\bar{u}\in \mathcal U$, we have

$$\begin{aligned} 0&\ge \big (\rho -\mathcal J'(u)\big )(\bar{u}-u)=\rho (\bar{u}-u)+\mathcal J'(u)(u-\bar{u})\\&\ge -\Vert \rho \Vert _{U^*}\Vert u-\bar{u}\Vert _{U} + c\Vert u-\bar{u}\Vert _{U}^{\mu +1}. \end{aligned}$$

Hence, $\Vert u-\bar{u}\Vert _{U}\le c^{-1/\mu }\Vert \rho \Vert _{U^*}^{1/\mu }.$ The result follows defining $\alpha :=\delta$ and $\kappa :=c^{-1/\mu }$.$\square$

Growth assumption (40) appeared first in [21, Assumption 2] as a natural hypothesis for an affine optimal control problem; see also [18, Proposition 4.3], where this kind of growth was proven for an elliptic optimal control problem under a linearized growth hypothesis. A similar assumption of this type appeared in [20, Assumption A2], where stability results for an affine optimal control problem were studied. In Proposition A.3 below, we give further details on growth (40) and its linearization.

Necessary conditions. In order to establish necessary conditions for stability in the form of growth properties of functionals, we will use Ekeland principle in the form of Lemma A.2, following the approach used in [3, 4]. In those papers, the subregularity property of the subdifferential of convex functions was characterized in terms of quadratic growth conditions; see also [33], where a similar approach was used for the limiting subdifferential. In all those three papers only Lipschitz stability and quadratic growth conditions were considered. We make simple refinements in those arguments to consider both Hölder stability and higher-order growth conditions.

In the next theorem, we argue similarly to the proof of [3, Theorem 3.3]; see also the proofs of [4, Theorem 2.1] and [33, Theorem 3.1] for parallel arguments.

Theorem A.2

Suppose $\big (U,\Vert \cdot \Vert _{U}\big )$ is a Banach space, $\mathcal U$ a closed convex subset of U and $\mathcal {J}:\mathcal U\rightarrow \mathbb R$ a lower semicontinuous Gateaux differentiable function. Let $\bar{u}\in \mathcal U$ be a local minimizer of $\mathcal {J}$ and $\mu \in (0,\infty )$. Suppose that the optimality mapping $\Phi :\mathcal U\twoheadrightarrow U^*$ is strongly Hölder subregular at $\bar{u}$ with exponent $1/\mu$. Then there exist positive numbers $\delta$ and c such that

$$\begin{aligned} \mathcal {J}(u)-\mathcal {J}(\bar{u})\ge c\Vert u-\bar{u}\Vert _{U}^{\mu +1}\quad \text {for all}\,u\in \mathcal U\hbox { with }\Vert u-\bar{u}\Vert _{U}\le \delta . \end{aligned}$$

(41)

Proof

Let $\alpha$ and $\kappa$ be positive numbers such that property (39) holds. Suppose that (41) does not hold. Then there would exist $u\in \mathcal U\setminus \{\bar{u}\}$ satisfying $\Vert u-\bar{u}\Vert _{U}<2\alpha /3$ such that

$$\begin{aligned} \mathcal {J}(u)-\mathcal {J}(\bar{u})<\frac{1}{2^{2\mu +1}\kappa ^{\mu }}\Vert u-\bar{u}\Vert _{U}^{\mu +1}. \end{aligned}$$

(42)

Let $\varepsilon :=2^{-(2\mu +1)}\kappa ^{-\mu }\Vert u-\bar{u}\Vert _{U}^{\mu +1}$ and $\lambda :=2^{-1}\Vert u-\bar{u}\Vert _{U}$. Note that

$$\begin{aligned} \lambda =\frac{3}{2}\Vert u-\bar{u}\Vert _{U}-\Vert u-\bar{u}\Vert _{U}<\alpha -\Vert u-\bar{u}\Vert _{U}. \end{aligned}$$

From Lemma A.2, we conclude the existence of $\hat{u}\in \mathcal U$ and ${\hat{\rho }}\in U^*$ such that

• (i) $\displaystyle \Vert u-\hat{u}\Vert _{U}\le \frac{1}{2}\Vert u-\bar{u}\Vert _{U}$;

• (ii) $\displaystyle \Vert {\hat{\rho }}\Vert _{U^*}\le \frac{1}{4^{\mu }\kappa ^\mu } \Vert u-\bar{u}\Vert _{U}^{\mu }$;

• (iii) ${\hat{\rho }}\in \mathcal {J}'(\hat{u})+N_\mathcal {U}(\hat{u})$.

Observe that $\Vert \hat{u}-\bar{u}\Vert _{U}\le \Vert \hat{u}-u\Vert _{U}+ \Vert u-\bar{u}\Vert _{U}\le 1/2\Vert u-\bar{u}\Vert _{U}+\Vert u-\bar{u}\Vert _{U}< \alpha$. By subregularity of the optimality mapping at $\bar{u}$, we get

$$\begin{aligned} \Vert \hat{u}-\bar{u}\Vert _{U}\le \kappa \Vert {\hat{\rho }}\Vert _{U^*}^{1/\mu }\le \frac{1}{4}\Vert u-\bar{u}\Vert _{U}. \end{aligned}$$

(43)

By item (i), we have $\Vert u-\bar{u}\Vert _{U}\le \Vert u-\hat{u}\Vert _{U}+\Vert \hat{u}-\bar{u}\Vert _{U}\le 1/2\Vert u-\bar{u}\Vert _{U}+\Vert \hat{u}-\bar{u}\Vert _{U}$. This implies $\Vert u-\bar{u}\Vert _{U}\le 2\Vert \hat{u}-\bar{u}\Vert _{U}$. Combining this with (43), we get

$$\begin{aligned} \Vert \hat{u}-\bar{u}\Vert _{U}\le \frac{1}{4}\Vert u-\bar{u}\Vert _{U}\le \frac{1}{2} \Vert \hat{u}-\bar{u}\Vert _{U}, \end{aligned}$$

and hence $\hat{u}=u=\bar{u}$. A contradiction to (42).$\square$

Growth condition (41) is well known in optimization. See, for example, [14, Theorem 2.4] or [31, Theorem III] in affine optimal control; and [30, Theorem 1] or [32, Theorem I] in the quantitative study of eigenvalues stability for the Schrödinger operator.

1.3 Hölder growth of real-valued functions

In this section, we study how to reduce growth conditions (40) and (41) to linearized versions. This is to facilitate the understanding of their feasibility. We say that $\mathcal J:\mathcal U\rightarrow \mathbb R$ has second variation at $\bar{u}\in \mathcal U$ if there exists a function $\mathcal J''(\bar{u}):U\times U\rightarrow \mathbb R$, positively homogeneous in each variable, such that

$$\begin{aligned} \mathcal {J''}(\bar{u})(v,w)=\lim _{\varepsilon \rightarrow 0^+}\frac{{\mathcal J'}(\bar{u}+\varepsilon v)w-\mathcal {J'}(\bar{u})w}{\varepsilon } \quad \forall v,w\in \mathcal U-\bar{u}. \end{aligned}$$

From now on, we will assume that $\mathcal {J}$ has second variation at every element of $\mathcal U$. We abbreviate $\mathcal J''(\bar{u})v^2:=\mathcal J''(\bar{u})(v,v)$.

A Hölder-type second order condition. In order to transfer conditions (40) and (41) from being satisfied by a nonlinear function to a second order polynomial, we will employ the following weakened version of “twice continuously differentiable".

Definition A.2

Let $\bar{u}\in \mathcal U$ and $\mu \in [1,\infty )$. We say that $\mathcal {J}$ has changing curvature of order $\mu$ at $\bar{u}$ if for every $\varepsilon >0$ there exists $\delta >0$ such that

$$\begin{aligned} |\mathcal J''(\bar{u}+v)v^2-\mathcal J''(\bar{u})v^2|\le \varepsilon \Vert v\Vert _{U}^{\mu +1} \end{aligned}$$

(44)

for all $v\in U$ with $\bar{u}+v\in \mathcal U$ and $\Vert v\Vert _{U}\le \delta$.

Properties like (44) have appeared ubiquitously in the optimal control literature of bang-bang controls. See, for example, [14, p. 4207], where it appeared as a standard assumption in abstract optimal control; or [19, Lemma 11] where it appeared as natural property in the context of parabolic optimal control problems.

Proposition A.1

Suppose that $\mathcal {J}$ has changing curvature of order $\mu \in [1,\infty )$ at $\bar{u}\in \mathcal U$, then the following statements hold:

• (i) For all $\varepsilon >0$ there exists $\delta >0$ such that

  $$\begin{aligned} |\mathcal {J}(\bar{u}+v)-\mathcal {J}(\bar{u})-\mathcal J'(\bar{u})v-\frac{1}{2}\mathcal J''(\bar{u})v^2|\le \varepsilon \Vert v\Vert _{U}^{\mu +1} \end{aligned}$$

  for all $v\in U$ satisfying $\bar{u}+v\in \mathcal U$ and $\Vert v\Vert _U\le \delta$.

• (ii) For all $\varepsilon >0$ there exists $\delta >0$ such that

  $$\begin{aligned} |\mathcal J'(\bar{u}+v)v-\mathcal J'(\bar{u})v-\mathcal J''(\bar{u})v^2|\le \varepsilon \Vert v\Vert _{U}^{\mu +1} \end{aligned}$$

  for all $v\in U$ satisfying $\bar{u}+v\in \mathcal U$ and $\Vert v\Vert _U\le \delta$.

Proof

For each $v\in U$ with $\bar{u}+v\in \mathcal U$, define $h_v:[0,1]\rightarrow \mathbb R$ by $h_v(t)=\mathcal {J}(\bar{u}+tv)$. We can apply the Taylor Theorem to conclude that for each $v\in \mathcal U-\bar{u}$ there exists $t_v\in [0,1]$ such that $h(1)-h(0)=h'(0)+h''(t_v)/2$. That is,

$$\begin{aligned} \mathcal {J}(\bar{u}+v)-\mathcal {J}(\bar{u})=\mathcal J'(\bar{u})v+\frac{1}{2}\mathcal J''(\bar{u}+t_v v)v^2\quad \forall v\in \mathcal U-\bar{u}. \end{aligned}$$

(45)

Let $\varepsilon >0$ be given. By definition of changing curvature of order $\mu$ at a point, we can find $\delta >0$ such that

$$\begin{aligned} |\mathcal J''(\bar{u}+t_v v)(t_v v)^2-\mathcal J''(\bar{u})(t_v v)^2|\le 2\, \varepsilon \Vert t_v v\Vert _{U}^{\mu +1} \end{aligned}$$

(46)

whenever $v\in \mathcal U-\bar{u}$ satisfies $\Vert v\Vert _{U}\le \delta$. Then, combining (45) and (46), we get

$$\begin{aligned}&t_{v}^{2}|\mathcal {J}(\bar{u}+v)-\mathcal {J}(\bar{u})-\mathcal J'(\bar{u})v-\frac{1}{2}\mathcal J''(\bar{u})v^2|\\&\quad =\frac{1}{2}|\mathcal J''(\bar{u}+t_v v)v^2-\mathcal J''(\bar{u})v^2|\le t_v^{\mu +1}\varepsilon \Vert v\Vert _{U}^{\mu +1} \end{aligned}$$

for all $v\in U$ satisfying $\bar{u}+v\in \mathcal U$ and $\Vert v\Vert _U\le \delta$. Since $\mu \ge 1$, it follows that

$$\begin{aligned} |\mathcal {J}(\bar{u}+v)-\mathcal {J}(\bar{u})-\mathcal J'(\bar{u})v-\frac{1}{2}\mathcal J''(\bar{u})v^2|\le t_{v}^{\mu -1}\varepsilon \Vert v\Vert _{U}^{\mu +1}\le \varepsilon \Vert v\Vert _{U}^{\mu +1} \end{aligned}$$

for all $v\in U$ satisfying $\bar{u}+v\in \mathcal U$ and $\Vert v\Vert _U\le \delta$. Thus, item (i) holds.

The proof of item (ii) is analogous; it follows defining $k_v:[0,1]\rightarrow \mathbb R$ given by $k_v(t):=\mathcal {J}'(\bar{u}+tv)v$ for each $v\in \mathcal U-\bar{u}$, and applying the Mean Value Theorem to each function $k_v$.$\square$

Growth of functionals and their differentials. One easy consequence of Proposition A.1 is the following characterization of growth condition (41) which follows directly from item (i) of Proposition A.1.

Proposition A.2

Suppose that $\mathcal {J}$ has changing curvature of order $\mu \in [1,\infty )$ at $\bar{u}\in \mathcal U$, then the following statements are equivalent:

• (i) There exist positive numbers $\alpha$ and c such that

  $$\begin{aligned} \mathcal {J}(u)-\mathcal {J}(\bar{u})\ge c\Vert u-\bar{u}\Vert _{U}^{\mu +1}\quad \text {for all}\,u\in \mathcal U\hbox { with }\Vert u-\bar{u}\Vert _{U}\le \alpha . \end{aligned}$$

• (ii) There exist positive numbers $\alpha$ and c such that

  $$\begin{aligned}&\mathcal J'(\bar{u})(u-\bar{u})+\frac{1}{2}\mathcal J''(\bar{u})(u-\bar{u})^2\\&\quad \ge c\Vert u-\bar{u}\Vert _{U}^{\mu +1}\quad \text {for all}\,u\in \mathcal U\hbox { with }\Vert u-\bar{u}\Vert _{U}\le \alpha . \end{aligned}$$

The growth in previous proposition has been proved in the literature of optimal control several times, this usually involves using a growth condition on the second variation over a critical cone and the so-called structural assumption; this is a condition one the level sets of the adjoint variable. See [6, Section 3] and [15, Theorem 2.4].

On the other hand, another trivial, but important, consequence of Proposition A.1 is the characterization of growth condition (40).

Proposition A.3

If $\mathcal {J}$ has changing curvature of order $\mu \in [1,\infty )$ at $\bar{u}\in \mathcal U$, then the following statements are equivalent.

• (i) There exist positive numbers $\alpha$ and c such that

  $$\begin{aligned} \mathcal J'(u)(u-\bar{u})\ge c\Vert u-\bar{u}\Vert _{U}^{\mu +1}\quad \text {for all}\,u\in \mathcal U\hbox { with }\Vert u-\bar{u}\Vert _{U}\le \alpha . \end{aligned}$$

• (ii) There exist positive numbers $\alpha$ and c such that

  $$\begin{aligned} \mathcal J'(\bar{u})v+\mathcal J''(\bar{u})v^2\ge c\Vert u-\bar{u}\Vert _{U}^{\mu +1}\quad \text {for all}\,u\in \mathcal U\hbox { with }\Vert u-\bar{u}\Vert _{U}\le \alpha . \end{aligned}$$

We mention that this characterization has appeared before in PDE-constrained optimization; see [18, Proposition 4.1] or [19, Lemma 12]. Numerous conditions exist to verify the validity of the growth condition given in Proposition A.3; see, e.g., [15, Lemma 2.5] or [18, Theorem 6.3]. A comprehensive discussion on the assumptions pertinent to this growth condition is available in [18, Section 6]. An explicit example where the growth holds for $\mu \in \mathbb N$ was given in [36, Example 1.2] (an optimal control problem constrained by ODEs).