3 Open loop optimal switching control
3.1 Existence of global minimizers
Suppose we take a switching signal q(·) ∈ Sess([0, T ]; Q) subject to our
specification in order to minimize a cost criterion of the form
J = ZT 0 J0(x(t, ·)) dt + ZT 0 J1(y(t)) dt+ K+1 X k=1 γ(τk− τk−1, x(τk, ·), qk−1yqk) + c T _ 0 q(t) δt (3.1.1)
where x(t, ·) is a solution of the switching system ∂ ∂tx(t, s) + A q(t)(t, s)∂ ∂sx(t, s) = f q(t)(t, s, x(t, s)), CqL(t)x|s=a = uqL(t), CqR(t)x|s=b = uqR(t), y(t) = CSx|s=b, x(0, s) = ¯x(s). (3.1.2)
for t ∈ [0, T ] with given initial and boundary data ¯x(·) ∈ BV((a, b); Rn), uqL(·) ∈ BV((0, T ); Rmq+) and uq
R(·) ∈ BV((0, T ); Rm
q
−) for all q ∈ Q.
Here J0(x(t, ·)) denotes costs for the solution x(t, ·) on (a, b) (e. g., costs for the distribution of material in a network at the time instance t), J1(y(·)) denotes costs for the output (e. g., a demand penalty) and γ(τk, x(τk, ·), qk−1 y qk) denotes switching costs, i. e., costs for switch-
where τk are the essential switching times as obtained in Lemma 1.5.1
and cWT
0 q(t) δt is a regularization term.
Note that J0(x(t, ·)) and J1(y(t)) are implicit functions of q(·) by the coupling in (3.1.2).
Our primary objective is to show the existence of an optimal switch- ing control q∗(·), i. e.,
{q∗(·) ∈ Sess: q∗(·) = argminq(·)∈Sess([0,T ];Q)J(q(·))} , ∅. (3.1.3)
We will impose the following hypotheses.
Hypotheses 3.1.1:
1.) The functions J0(·) : L1((a, b); Rn) → R and J1(·) : Rns → R are
non-negative and lower-semicontinuous.
2.) For all q, q′ ∈ Q, the function γ(·, ·, q y q′) : R× L1((a, b); Rn)→
Ris non-negative, continuous in the first component, lower-semi- continuous in the second component and satisfies γ(0, x, q y q′) = 0 for all x ∈ L1((a, b); Rn).
3.) The regularization parameter c ∈ R is strictly positive.
4.) For some ¯q(·) ∈ Sess([0, T ]; Q), the costs J( ¯q) are finite.
Observe that by Hypothesis 3.1.1.2, the switching costs of signals in q(·) ∈ Sess((0, T ); Q) coincide with the costs of any pointwise rep-
resentative in ¯q(·) ∈ S([0, T ]; Q) possibly involving cascaded switches q yy q′′... ′ at the essential switching times τk (or loops q yy q at a
subset of measure zero in [0, T ]).
We then get a general existence result for this optimal control prob- lem.
Theorem 3.1.1. Assume the Hypotheses 2.0.1, 2.0.2, 2.0.3 and 3.1.1. Then, for T > 0, there exists an optimal q∗(·) ∈ S
ess([0, T ]; Q) with at most
K = J( ¯q(·)) c
(3.1.4)
3.1 Existence of global minimizers Proof. Let
J∗= inf
q(·)∈Sess((0,T );Q)
J(q(·)). (3.1.5)
Due to Hypothesis 3.1.1.4, there exists a minimizing sequence {qν(·)}ν∈N
in Sess((0, T ); Q) such that
lim
ν→∞J(q
ν(·) = J∗. (3.1.6)
Now assume that there exists an optimal ˜q∗(·) with more than K es- sential switches and K given by (3.1.4). Under the Hypotheses 3.1.1.3 and 3.1.1.4, the corresponding optimal value then satisfies J( ˜q∗) > Kc, because the variation of ˜q∗(·) is increased at least by one with every essential switch. But, from the bound (3.1.4) we have J( ¯q) = Kc, con- tradicting the optimality of ˜q∗(·). Hence, (3.1.4) bounds the number of switches for any optimal q(·).
Then observe that (3.1.4) also bounds the variation WT
0 q∗(t) δt and
that R0T|q(t)| dt 6 T |Q|, so we can apply Theorem A.1.4 and, by this compactness, extract a subsequence (relabeled by ν) such that qν(·) ⇀
q∗(·) in Sess((0, T ); Q). By Proposition 2.0.2, the corresponding solutions
xν(t, ·) weakly converge to x∗(t, ·) in BV((a, b); Rn) and thus strongly
in L1((a, b); Rn) for almost every t ∈ (0, T ).
The lower semicontinuity assumptions on the functions J0 and γ in
Hypotheses 3.1.1.1 and 3.1.1.2 and the lower semicontinuity of the vari- ation imply a similar lower semicontinuity of J
J(q∗(·)) 6 lim inf
ν→∞ J(q
ν(·)) = J∗, (3.1.7)
so the minimum J∗ is attained at q∗(·).
Remark 3.1.1: One could also consider a corresponding infinite horizon
problem T → ∞, for which one would include a discount e−λtin (3.1.1), so minimizing J∞= Z∞ 0 J0(x(t, ·))e−λtdt + Z∞ 0 J1(y(t))e−λtdt + K+1 X k=1 γ(τk− τk−1, x(τk, ·), qk−1yqk)e−λτk + c ∞ _ 0 q(t)e−λtδt, (3.1.8)
cf. also Remark 1.5.1. Finiteness of J∞ may then depend on having a
large enough discount rate λ > 0 in (3.1.8). Assuming Hypothesis 3.1.1 on every bounded subinterval [0, T ], there exists a minimizing sequence for J∞. From this sequence, one can, as in the proof of Theorem 3.1.1,
extract a subsequence convergent on [0, 2], [0, 3], . . . stepwise and, by a diagonal argument, obtain a minimizing sequence qν(·) weakly conver- gent to q∗(·) in Sess([0, T ]; Q) for every [0, T ]. Since qν is a minimizing
sequence, we have
J∞(qν(·)) 6 J∞
∗ + εν with εν→ 0, (3.1.9)
and, for each T we have (again as in the proof of Theorem 3.1.1, noting that each {qν(·)} converges on [0, T ])
JT(q∗(·)) 6 JT(qν(·)) + εν(T ) with εν(T )→ 0. (3.1.10) Then, for each T ,
JT(q∗(·)) 6 JT(qν(·)) + ε(T ) 6 J∞(qν(·)) + εν(T ) 6 J∞
∗ + εν(T ) + εν.
Letting T → ∞ gives J∞(q∗(·)) 6 J∞
∗ , so this limit switching function
q∗(·) ∈ Sess((0, ∞); Q) then minimizes J∞.
Remarks on combinatorial constraints
For the existence result in Theorem 3.1.1, we assumed unconstraint switching, i. e., the modal transition from each q ∈ Q to any other mode q′∈ Q is always feasible (so the corresponding graph with edges q y q′ in Figure 1.1 is complete). Eventual combinatorial constraints, for example of the form that a transition ˆq y ˆq′y ˆq′′for some ˆq, ˆq′, ˆq′′
is feasible, but a direct switch ˆq y ˆq′′ is forbidden, are not inherently covered in this setting and must be taken into account by an appropriate penalization in the cost function. In order to treat constraints of latter form, we might for instance require that the switching costs satisfy
γ(·, ·, ˆq y ˆq′′) > γ(·, ·, ˆq y ˆq′) + γ(·, ·, ˆq′y ˆq′′). (3.1.11) It shall be remarked that the theory for BV based on essential switch- ing signals q(·) ∈ Sess([0, T ]; Q) at this point differs from the theory
3.2 A characterization of optimal switching boundary control
for the scalar equation [27] using pointwise switching signals q(·) ∈ S([0, T ]; Q) and piecewise continuous functions in Cpw([a, b]) as intro-
duced in Section 1.4. There, the regularization term cWT
0 q(t) δt is not
needed and a bound on the number of switches K can be achieved by bounding γ(·, ·, q y q′) away from 0. Eventual combinatorial con- straints as those above are then inherently covered, but with accepting the possibility of a cascaded switch ˆq y ˆq′ y ˆq′′ at the same time as feasible.