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.
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Acknowledgments
The author is grateful to the anonymous referees for their instructive comments which, in particular, helped to close a gap in one of the main proofs.
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Appendix
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
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
and analogous formulas for \(w_{j-k}\), it follows from the assumptions on w that
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,
(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
For part (i) we thus obtain by induction
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 \(u_{i}=\tilde{u}^{h}_{i}\):
due to the construction of \(\tilde{u}^{h}\). Therefore,
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 \(i=0,\dots ,k\),
where \(\sigma ^{\tau }\) abbreviates \(\sigma ^{\tau }(t)=\sigma (t+\tau _{s}-t_{k})\). Consequently,
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
For \(t\le \tau _{s}\),
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
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
Thus,
Taking into account assumption (H0), from Theorem 1 and Lemma 2 conclude
In analogy to Lemma 7, the boundary terms are equally estimated by \(\hbox {O}(h)\). Using Lemma 14 we see that
and the estimate for \(\Vert \hat{\sigma }-\sigma ^{0}\Vert _{\infty }\) directly follows.
Consider next \(\Delta ^{1}(\hat{\sigma }-\sigma ^{0})\):
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 \(\Delta ^{2}(\hat{\sigma }-\sigma ^{0})\):
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
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 \(r\in \{2,\infty \}\). \(\square \)
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Felgenhauer, U. Discretization of semilinear bang-singular-bang control problems. Comput Optim Appl 64, 295–326 (2016). https://doi.org/10.1007/s10589-015-9800-2
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DOI: https://doi.org/10.1007/s10589-015-9800-2