Discretization of semilinear bang-singular-bang control problems

Abstract

Bang-singular controls may appear in optimal control problems where the control enters the system linearly. We analyze a discretization of the first-order system of necessary optimality conditions written in terms of a variational inequality (or: inclusion) under appropriate assumptions including second-order optimality conditions. For the so-called semilinear case, it is proved that the discrete control has the same principal bang-singular-bang structure as the reference control and, in $L_{1}$ topology, the convergence is of order one w.r.t. the stepsize.

Appendix

In this section, we first derive several norm estimates for functions from $W_{2}^{1}$ resp. their discrete analoga from $Y^{h}$. Afterwards, the proofs of Lemmas 2 and 10 are given.

Lemma 13

Let $w\in \mathbb {R}^{k(N+1)}$ with $\Delta ^{1}w\in \mathbb {R}^{kN}$ be such that

$$\begin{aligned} \Vert w\Vert _{\infty }+\,\Vert \Delta ^{1}w\Vert _{2}\le \,M_0\,<\,\infty . \end{aligned}$$

Then,  $\Vert w\Vert ^2_{\infty }\le \,M\,(h\,+\,\Vert w\Vert _{(2)})$  for some $M>0$ independent of $h,\,w$.

Proof

Let j be an index where $|w_{j}|=\Vert w\Vert _{\infty }$, and c a constant greater than $M_0$. Using

$$\begin{aligned} w_{j+k}\,=\,w_{j}+\sum _{l=0}^{k-1}h(\Delta ^1 w)_{j+l} \end{aligned}$$

and analogous formulas for $w_{j-k}$, it follows from the assumptions on w that

$$\begin{aligned} |w_{i}|\,\ge \,|w_{j}|\,-\,c\sqrt{h\,|i-j|}\, \ge \, |w_{j}|\left( 1-\frac{1}{\sqrt{2}}\right) \end{aligned}$$

for all i such that $|i-j|\le m:=\lfloor |w_{j}|^2\cdot {N}/(2c^2)\rfloor$ and $t_{i}\in [0,1]$. Taking into account $|w_{j}|^2\cdot {N}/(2c^2)\le M_0^2\,N/(2c^2)< N/2$,  it is easy to see that the conditions are fulfilled at least for $j<i\le j+m$ in case $t_{j}\le 1/2$  (or $j-m\le i<j$ for $t_{j}> 1/2$) , i.e. on some index set I(w) containing at least m knots on [0, 1].  Consequently,

$$\begin{aligned} \Vert w\Vert _{(2)}^{2}\ge & {} h\,\sum _{i\in I(w)}|w_{i}|^{2}\,\ge \,\frac{mh}{2}(1-\sqrt{2})^{2}\,\Vert w\Vert _{\infty }^{2}\\\ge & {} \frac{h}{2}(3-2\sqrt{2})\,\Vert w\Vert _{\infty }^{2} \left( \frac{|w_{j}|^2\cdot N}{2c^2}-\,1\right) \,=\,2\,M_1\Vert w\Vert _{\infty }^{2}\left( \Vert w\Vert ^2_{\infty }-\,2c^2h\right) \end{aligned}$$

(with $M_1=(3-2\sqrt{2})/(8c^2)$)  due to $h\cdot N=1$.

If now   $\Vert w\Vert ^2_{\infty }\ge \,4hc^2$ then $\Vert w\Vert _{(2)}^{2}\ge \,M_1\Vert w\Vert _{\infty }^{4}$. Otherwise, $\Vert w\Vert ^2_{\infty }< \,4hc^2$ and hence the lemma. $\square$

Remark  For continuous functions $w\in W_{2}^{1}(0,1;\mathbb {R}^{k})$ with $\Vert w\Vert _{\infty }+\,\Vert \dot{w}\Vert _2\le M_0$, similarly get   $\Vert w\Vert _{\infty }\le \,M\,\Vert w\Vert _{2}^{1/2}$ where M depends only on $M_0$.

The next lemma is a special case of the discrete analogon of Gronwall’s Lemma [22]. In order to emphasize the independence of related constants of the step size h, a short direct proof is provided.

Lemma 14

(Discrete Gronwall Lemma) Suppose there are given an arbitrary $\phi \in \mathbb {R}^{kN}$ and some constant $L>0$.

• (i) If  $\eta \in \mathbb {R}^{k(N+1)}$ satisfies $|(\Delta ^{1}\eta )_{i}|\le L\,|\eta _{i}|+|\phi _{i}|$ for $i=0,\dots ,N-1$, then   $\Vert \eta \Vert _{\infty }\le \;e^{L}(|\eta _{0}|\,+\,\Vert \phi \Vert _{2})$.

• (ii) If  $\eta \in \mathbb {R}^{k(N+1)}$ satisfies $|(\Delta ^{1}\eta )_{i}|\le L\,|\eta _{i+1}|+|\phi _{i}|$ for $i=0,\dots ,N-1$, then   $\Vert \eta \Vert _{\infty }\le \;e^{L}(|\eta _{N}|\,+\,\Vert \phi \Vert _{2})$.

Proof

The proof starts with the observation

$$\begin{aligned} \left| \,|\eta _{i+1}|-|\eta _{i}|\,\right| \,\le \,h\,|(\Delta ^{1}\eta )_{i}|. \end{aligned}$$

For part (i) we thus obtain by induction

$$\begin{aligned} |\eta _{i+1}|\le & {} (1+Lh)|\eta _{i}|\,+\,h\,|\phi _{i}| \;\le \; (1+Lh)^{i+1}|\eta _{0}|\,+\,h\sum _{j=0}^{i}(1+Lh)^{i-j}|\phi _{j}|\\\le & {} (1+Lh)^{N}\left( |\eta _{0}|\,+\,h\sum _{j=0}^{N-1}|\phi _{j}|\right) \;\le \;e^{L}\left( |\eta _{0}|\,+\,\Vert \phi \Vert _{2}\right) . \end{aligned}$$

Similarly, part (ii) follows from  $|\eta _{i}|\le (1+Lh)|\eta _{i+1}|+h|\phi _{i}|,\;i\le N-1$. $\square$

Proof of Lemma 2

Consider first $x_{i}=\tilde{x}^{h}_{i},\,i=0,\dots ,N$, and $u_{i}=\tilde{u}^{h}_{i}$:

$$\begin{aligned} x_{i+1}-x_{i}= & {} \int _{\omega _{i}}\dot{x}^{0}(t)\,dt\,=\, \int _{\omega _{i}}\left[ f(x^{0}(t))\,+\,B\,u^{0}(t)\right] \,dt\\= & {} \int _{\omega _{i}}\left[ f(x^{0}(t))-f(x^{0}(t_{i}))\right] \,dt\; +\,h\left[ f(x_{i})+B\,u_{i}\right] \end{aligned}$$

due to the construction of $\tilde{u}^{h}$. Therefore,

$$\begin{aligned} \tilde{\delta }^{h}_{1i} =\,h^{-1}\int _{t_{i}}^{t_{i}+h}\left[ f(x^{0}(t)) -f(x^{0}(t_{i}))\right] \,dt \,=\;\text{ O }(h). \end{aligned}$$

Analogous estimates show that $\tilde{\delta }^{h}_{2i}=\hbox {O}(h)$ uniformly for $i=0,\dots ,N-1$.

The construction of $\tilde{\mu }^{h}_{i}$ ensures (19) to be valid for each $i\le N-1$ if only h is sufficiently small. In order to find $\tilde{\delta }^{h}_{3}$, insert $\tilde{x}^{h}_{i}, \tilde{p}^{h}_{i}$ and $\tilde{\mu }^{h}_{i}$ into (18): for $i=0,\dots ,k$,

$$\begin{aligned} B^{T}p_{i}=\,\sigma ^{0}(t_{i})\; =\;\left\{ \begin{array}{llcl} \tilde{\sigma }^{h}_{i}&{}+\;(\sigma ^{0}(t_{i})-\sigma ^{\tau }(t_{i})) &{}\quad \text{ if }&{}\quad i\le k,\\ \tilde{\sigma }^{h}_{i}&{}&{}\quad \text{ if }&{}\quad i>k. \end{array}\right. \end{aligned}$$

where $\sigma ^{\tau }$ abbreviates $\sigma ^{\tau }(t)=\sigma (t+\tau _{s}-t_{k})$. Consequently,

$$\begin{aligned} \left| \tilde{\delta }^{h}_{3i}\right| =|\sigma ^{0}(t_{i}) -\sigma ^{\tau }(t_{i})| \,=\,\text{ O }(h) \qquad \text{ for }\;i\le k, \end{aligned}$$

and $\tilde{\delta }^{h}_{3i}=0$ in case $i>k$.   For $\Delta ^{1}\tilde{\delta }^{h}_{3}, \,\Delta ^{2}\tilde{\delta }^{h}_{3}$ we have

$$\begin{aligned} (\Delta ^{1}\tilde{\delta }^{h}_{3})_{i}= & {} \left\{ \begin{array}{lcl} (\Delta ^{1}\sigma ^{0})_{i}-\,(\Delta ^{1}\sigma ^{\tau })_{i}&{}\text{ if }&{}\quad i\le k-1,\\ -h^{-1}\sigma ^{0}(t_{k}) &{}\text{ if }&{}\quad i= k,\\ \;0&{}\text{ if }&{}\quad i> k, \end{array}\right. \\ (\Delta ^{2}\tilde{\delta }^{h}_{3})_{i}= & {} \left\{ \begin{array}{lcl} (\Delta ^{2}\sigma ^{0})_{i}-\,(\Delta ^{2}\sigma ^{\tau })_{i}&{}\text{ if }&{}\quad i\le k-1,\\ h^{-2}(\sigma ^{0}(t_{k-1})-\sigma ^{\tau }(t_{k-1})-2\sigma ^{0}(t_{k})) &{}\text{ if }&{}\quad i= k,\\ h^{-2}\sigma ^{0}(t_{k})&{}\text{ if }&{}\quad i= k+1,\\ \;\,0&{}\text{ if }&{}\quad i\ge k+2. \end{array}\right. \end{aligned}$$

For $t\le \tau _{s}$,

$$\begin{aligned} \sigma ^{0}(t)\, =\int _{\tau _{s}}^{t}\int _{\tau _{s}}^{s}\ddot{\sigma }^{0}(\theta )\,d\theta \,ds\, =\,\text{ O }((\tau _{s}-t)^{2}) \end{aligned}$$

so that $|\sigma ^{0}(t_{k})|+|\sigma ^{0}(t_{k-1})|=\hbox {O}(h^{2})$. Analogously obtain $|\sigma ^{\tau }(t_{k-1})|=\hbox {O}(h^{2})$, too. Together with the estimates

$$\begin{aligned} |(\Delta ^{1}\sigma ^{\tau })_{i}-\,(\Delta ^{1}\sigma ^{0})_{i}|+ |(\Delta ^{2}\sigma ^{\tau })_{i}-\,(\Delta ^{2}\sigma ^{0})_{i}|= \text{ O }(h), \end{aligned}$$

the desired results for $\tilde{\delta }_{3}^{h}$ and its finite differences directly follow. $\square$

Proof of Lemma 10

By definition, $\hat{\sigma }=B^{T}\hat{p}$ where $\hat{p}$ solves the backward initial value problem for the finite difference equation

$$\begin{aligned} (\Delta ^{1}\hat{p})_{i}=\,-\nabla f(\hat{x}_{i+1})^{T}\hat{p}_{i+1},\qquad \hat{p}_{N}=\,-\nabla k(\hat{x}_{N}). \end{aligned}$$

Thus,

$$\begin{aligned} (\Delta ^{1}(\hat{p}-p^{0}))_{i}+\,A_{i+1}(\hat{p}-p^{0})_{i+1}= & {} \left[ \nabla f(x^{0}_{i+1})-\nabla f(\hat{x}_{i+1})\right] ^{T}\hat{p}_{i+1}-\,\tilde{\delta }^{h}_{2i},\\ \hat{p}_{N}-\,p^{0}_{N}= & {} \nabla k(x^{0}_{N})-\nabla k(\hat{x}_{N}). \end{aligned}$$

Taking into account assumption (H0), from Theorem 1 and Lemma 2 conclude

$$\begin{aligned} \Vert \hat{x}-x^{0}\Vert _{2}+\,\Vert \hat{p}- p^{0}\Vert _{2}=\,\text{ O }(\Vert \tilde{\delta }^{h}\Vert _{D}) \,=\,\text{ O }(h). \end{aligned}$$

In analogy to Lemma 7, the boundary terms are equally estimated by $\hbox {O}(h)$. Using Lemma 14 we see that

$$\begin{aligned} \Vert \hat{p}-p^{0}\Vert _{\infty }=\,\text{ O }(h) \end{aligned}$$

(62)

and the estimate for $\Vert \hat{\sigma }-\sigma ^{0}\Vert _{\infty }$ directly follows.

Consider next $\Delta ^{1}(\hat{\sigma }-\sigma ^{0})$:

$$\begin{aligned} (\Delta ^{1}(\hat{\sigma }-\sigma ^{0}))_{i}= & {} B^{T} (\Delta ^{1}(\hat{p}-p^{0}))_{i}\\= & {} -B^{T}\left[ (\nabla \hat{f}_{i+1})^{T}\hat{p}_{i+1}-(\nabla f^{0}_{i+1})^{T}p^{0}_{i+1}+\tilde{\delta }^{h}_{2i}\right] \\= & {} \text{ O }(|\hat{x}_{i+1}-x^{0}_{i+1}|+|\hat{p}_{i+1}-p^{0}_{i+1}| +|\tilde{\delta }^{h}_{2i}|) \end{aligned}$$

so that $\Vert \Delta ^{1}(\hat{\sigma }-\sigma ^{0})\Vert _{2}=\hbox {O}(h)$ follows from Theorem 1 and Lemma 2.

It remains to find a representation and estimates for $\Delta ^{2}(\hat{\sigma }-\sigma ^{0})$:

$$\begin{aligned} (\Delta ^{2}\hat{\sigma })_{i}= & {} h^{-1}B^{T}\left( (\Delta ^{1}\hat{p})_{i}- (\Delta ^{1}\hat{p})_{i-1}\right) \\= & {} -B^{T} \left( (\Delta ^{1}[\nabla \hat{f}])_{i}^{T}\hat{p}_{i+1} +\nabla \hat{f}_{i}^{T}(\Delta ^{1}\hat{p})_{i}\right) \\= & {} B^{T}\nabla \hat{f}_{i}^{T}\nabla \hat{f}_{i+1}^{T}\hat{p}_{i+1}-\, B^{T}h^{-1}\int _{\omega _{i}}\nabla _{xx}^{2} (\hat{p}_{i+1}^{T}\hat{f}[t])\,dt\cdot (\Delta ^{1}\hat{x})_{i}\\=: & {} \hat{P}_{i}+ \hat{R}_{i}\hat{u}_{i}. \end{aligned}$$

where $\hat{f}[t]=f(\hat{x}(t))$, and $\hat{x}(t)$ stands for the linear interpolation to $\hat{x}_{i}, \hat{x}_{i+1}$ on $\omega _{i}$. Therefore, the formula

$$\begin{aligned} \hat{R}_{i}-\,R_{i}=\, -B^{T}h^{-1}\int _{\omega _{i}}\left( \nabla _{xx}^{2}(\hat{p}_{i+1}^{T}\hat{f}[t])-\nabla _{xx}^{2}((p^{0})^{T}f^{0})_{i} \right) dt\cdot B \end{aligned}$$

leads to the desired estimates for the (discrete) $L_{2}$ and $L_{\infty }$ norms. Similarly, $\Vert \hat{P}-P\Vert _{r}$ can be estimated for $r\in \{2,\infty \}$. $\square$