Time optimal control problems: Bang-bang property and observability - - PowerPoint PPT Presentation
Time optimal control problems: Bang-bang property and observability estimates from measurable sets Can ZHANG (zhangcansx@163.com) Paris, June 2016 In honor of Prof. J. M. Corons 60th birthday occasion C. ZHANG Time optimal control problems
Bang-bang property and observability estimates from measurable sets
Can ZHANG (zhangcansx@163.com) Paris, June 2016 In honor of Prof. J. M. Coron’s 60th birthday occasion
Time optimal control problems 1 / 21
1
Motivations Observability estimates over measurable sets implies the bang-bang property
2
Main results
3
Conclusion and further research
Time optimal control problems 2 / 21
Bang-bang property of time optimal controls We introduce the time optimal control problem for the heat equation. Let ω ⊂ Ω be an open subset with its characteristic function χω. Consider the time optimal control problem: (TP)M
1 :
T(M) inf
u∈UM
Time optimal control problems 2 / 21
Bang-bang property of time optimal controls We introduce the time optimal control problem for the heat equation. Let ω ⊂ Ω be an open subset with its characteristic function χω. Consider the time optimal control problem: (TP)M
1 :
T(M) inf
u∈UM
where UM is the control constraint given by UM =
, M > 0, and y(·; u) solves the non-homogeneous heat equation yt − ∆y = χωu in Ω × R+, y = 0
y(x, 0) = y0(x) in Ω.
Time optimal control problems 2 / 21
control if y(T(M); u∗) = 0.
Time optimal control problems 3 / 21
control if y(T(M); u∗) = 0. In fact, the existence of optimal time and time optimal controls are guaranteed by the observability inequality and the decay of energy of the heat equation.
Time optimal control problems 3 / 21
control if y(T(M); u∗) = 0. In fact, the existence of optimal time and time optimal controls are guaranteed by the observability inequality and the decay of energy of the heat equation.
has no interior point in L2(Ω), to the best of our knowledge, we do not know how to separate this set from the target set {0} by a hyperplane in L2(Ω). Thus, we do not know how to get the Pontryagin maximum principle for Problem (TP)M
1 by the way used in the case of O.D.E..
Time optimal control problems 3 / 21
control if y(T(M); u∗) = 0. In fact, the existence of optimal time and time optimal controls are guaranteed by the observability inequality and the decay of energy of the heat equation.
has no interior point in L2(Ω), to the best of our knowledge, we do not know how to separate this set from the target set {0} by a hyperplane in L2(Ω). Thus, we do not know how to get the Pontryagin maximum principle for Problem (TP)M
1 by the way used in the case of O.D.E..
for (TP)M
1 : Any time optimal control u∗ for (TP)M 1 verifies |u∗(x, t)| = M
for a.e. (x, t) ∈ ω × (0, T(M)).
Time optimal control problems 3 / 21
The advantage of B-B-P
The argument is very simple. Assume u∗ and v∗ are two time optimal
2
= |u∗(x, t)|2 + |v∗(x, t)|2 2 −
2
, and (u∗ + v∗)/2 is also a time optimal control, by the B-B-P, we get that u∗ = v∗ a.e. in ω × (0, T(M)).
Time optimal control problems 4 / 21
The advantage of B-B-P
The argument is very simple. Assume u∗ and v∗ are two time optimal
2
= |u∗(x, t)|2 + |v∗(x, t)|2 2 −
2
, and (u∗ + v∗)/2 is also a time optimal control, by the B-B-P, we get that u∗ = v∗ a.e. in ω × (0, T(M)).
condition for the time optimal control problem.
controls for internally controlled heat equations, SICON, 2012.
Time optimal control problems 4 / 21
Observability inequalities from measurable sets To present our motivations, we begin with the simplest situation. Let T > 0 and Ω be a bounded Lipschitz domain in Rn. Consider the heat equation ∂tu − ∆u = 0, in Ω × (0, T), u = 0,
u(·, 0) = u0 ∈ L2(Ω). Two important a priori estimates for the above equation are as follows. Interior case: u(T)L2(Ω) ≤ N(Ω, T, D)
|u(x, t)| dxdt, ∀ u0 ∈ L2(Ω), (1) where D is a measurable subset of Ω × (0, T).
Time optimal control problems 5 / 21
Boundary case: u(T)L2(Ω) ≤ N(Ω, T, J )
| ∂
∂ν u(x, t)| dσdt, ∀ u0 ∈ L2(Ω),
(2) where J is a measurable subset of ∂Ω × (0, T).
Time optimal control problems 6 / 21
Boundary case: u(T)L2(Ω) ≤ N(Ω, T, J )
| ∂
∂ν u(x, t)| dσdt, ∀ u0 ∈ L2(Ω),
(2) where J is a measurable subset of ∂Ω × (0, T).
Control Theory when the observation regions are open subsets.
Time optimal control problems 6 / 21
Boundary case: u(T)L2(Ω) ≤ N(Ω, T, J )
| ∂
∂ν u(x, t)| dσdt, ∀ u0 ∈ L2(Ω),
(2) where J is a measurable subset of ∂Ω × (0, T).
Control Theory when the observation regions are open subsets. Our aim is to build up estimates (1) and (2) when D and J are subsets of positive measure and positive surface measure, respectively.
Time optimal control problems 6 / 21
Observability estimates from measurable sets implies the B-B-P Main idea: By contradiction, we would suppose that there were a constant ε ∈ (0, M) and a subset of positive measure D ⊂ ω × (0, T(M)) such that |u∗(x, t)| ≤ M − ε, ∀ (x, t) ∈ D. It provides a “room” for constructing another control (by the duality) such that there exist δ ∈ (0, T(M)) and v ∈ L∞(Ω × R+), with vL∞ ≤ M, such that ∂ty − ∆y = χωv in Ω × (0, T ∗ − δ), y = 0
∂Ω × (0, T ∗ − δ), y(x, 0) = y0(x) in Ω, y(x, T ∗ − δ) = 0 in Ω. This leads to a contradiction with the time optimality of T(M).
boundary control of the heat equation, SICON, 1997.
for the time optimal control problem, SICON, 2008.
Time optimal control problems 7 / 21
1
Motivations
2
Main results Heat equations Abstract evolution equations
An specific example
3
Conclusion and further research
Time optimal control problems 8 / 21
Recall the following result in the last talk given by M. Santiago:
Theorem
Assume that △4R(q0) (when non-empty) is real-analytic. Then, there are constants N and ρ, with 0 < ρ ≤ 1, such that |∂α
x ∂β t et∆f (x)| ≤ N (t − s)− n
4 e8R2/(t−s)|α|! β!
(Rρ)|α| ((t − s) /4)β es∆f L2(Ω), when x ∈ B2R(q0) ∩ Ω, 0 ≤ s < t, α ∈ Nn and β ≥ 0. Here △4R(q0) B4R(q0) ∩ ∂Ω.
Time optimal control problems 8 / 21
Theorem
Assume that f : B2R ⊂ Rn − → R is real-analytic in B2R verifying |∂α
x f (x)| ≤ M|α|!
(ρR)|α| , when x ∈ B2R, α ∈ Nn, for some M > 0 and 0 < ρ ≤ 1. Let E ⊂ BR be a measurable set with positive measure. Then, there are positive constants N = N(ρ, |E|/|BR|) and θ = θ(ρ, |E|/|BR|), with θ ∈ (0, 1), such that f L∞(BR) ≤ N
|f | dx θ M1−θ.
problem, Forum Math., 11 (1999), 695–703.
Time optimal control problems 9 / 21
Main results Based on these two theorems and a telescoping series method introduced in the last talk, we have The observability estimates (1) and (2) over measurable sets of positive measure in both space and time variables are true.
Time optimal control problems 10 / 21
Main results Based on these two theorems and a telescoping series method introduced in the last talk, we have The observability estimates (1) and (2) over measurable sets of positive measure in both space and time variables are true. Consequently, (TP)M
1 has B-B-P, i.e., any time optimal control u∗ satisfies
|u∗(x, t)| = M for a.e. (x, t) ∈ ω × (0, T(M)).
Time optimal control problems 10 / 21
Main results Based on these two theorems and a telescoping series method introduced in the last talk, we have The observability estimates (1) and (2) over measurable sets of positive measure in both space and time variables are true. Consequently, (TP)M
1 has B-B-P, i.e., any time optimal control u∗ satisfies
|u∗(x, t)| = M for a.e. (x, t) ∈ ω × (0, T(M)).
and obtain the corresponding bang-bang property.
measurable sets, J. Eur. Math. Soc., 16 (2014) 2433-2475.
Time optimal control problems 10 / 21
Extensions These observability estimates from measurable subsets can also be extended to solutions of the general higher-order parabolic equations with both space and time analytic coefficients.
parabolic analytic evolutions and applications, J. Math. Pures Appl., 104 (2015) 837-867.
evolutions and applications. Submitted.
Time optimal control problems 11 / 21
Abstract time optimal control problem X and U are two Hilbert spaces A generate a C0 semigroup {S(t); t ≥ 0} on X B ∈ L(U, X−1) is an admissible control operator for {S(t); t ≥ 0}, where X−1 is the dual of D(A∗) with respect to the pivot space X The controlled equation reads: dz dt = Az + Bf , t > 0, z(0) = z0 ∈ X, f ∈ L2
loc(R+; U).
(3) The time optimal control problem is as (TP)M
2 :
T(M) inf
f ∈UM
where z1 ∈ X is the target which differs from z0 and UM =
with M > 0.
Time optimal control problems 12 / 21
In this problem, T(M) is called the optimal time (if it exists), f ∗ ∈ UM is called an optimal control if z(T(M); f ∗, 0, z0) = z1.
Theorem
Let A generate an analytic semigroup {S(t); t ≥ 0} in X. Assume that (A∗, B∗) satisfies the observability inequality from time intervals: S(L)∗ϕ02
X ≤ e
d Lk
L B∗S(t)∗ϕ02
U dt,
for all ϕ0 ∈ D(A∗) and L ∈ (0, 1], where positive constants d and k are independent of L and ϕ0. Then the problem (TP)M
2 holds the bang-bang
property.
Time optimal control problems 13 / 21
In this problem, T(M) is called the optimal time (if it exists), f ∗ ∈ UM is called an optimal control if z(T(M); f ∗, 0, z0) = z1.
Theorem
Let A generate an analytic semigroup {S(t); t ≥ 0} in X. Assume that (A∗, B∗) satisfies the observability inequality from time intervals: S(L)∗ϕ02
X ≤ e
d Lk
L B∗S(t)∗ϕ02
U dt,
for all ϕ0 ∈ D(A∗) and L ∈ (0, 1], where positive constants d and k are independent of L and ϕ0. Then the problem (TP)M
2 holds the bang-bang
property. Here the bang-bang property means that: any optimal control f ∗ satisfies f ∗(t)U = M for a.e. t ∈ (0, T(M)).
for some evolution equations. Submitted.
Time optimal control problems 13 / 21
An Specific Example: Time-optimal controlled Stokes equations Assume that Ω ⊂ R3 is a bounded domain with a smooth boundary ∂Ω. Consider the controlled Stokes system yt − ∆y + ∇p = f , in Ω × R+, div y = 0,
y = 0, in ∂Ω × R+, y(·, 0) = y0, in Ω, (4) where y0 is arbitrarily fixed in the usual space: L2
σ(Ω) {y ∈ (L2(Ω))3 : div y = 0, y · ν = 0 on ∂Ω},
and f is taken from the control constraint set: UM
with M > 0, ω ⊂ Ω a nonempty open subset.
Time optimal control problems 14 / 21
The time optimal control problem now reads: (TP)M
3 :
T(M) inf
f ∈UM
Time optimal control problems 15 / 21
The time optimal control problem now reads: (TP)M
3 :
T(M) inf
f ∈UM
To apply the above-mentioned abstract result to obtain the bang-bang property for (TP)M
3 , we write X = L2 σ(Ω) and U = {0} × L2(ω) × L2(ω).
Time optimal control problems 15 / 21
The time optimal control problem now reads: (TP)M
3 :
T(M) inf
f ∈UM
To apply the above-mentioned abstract result to obtain the bang-bang property for (TP)M
3 , we write X = L2 σ(Ω) and U = {0} × L2(ω) × L2(ω).
Define the operator A on X by
0(Ω)
3 L2
σ(Ω),
Ay = P(∆y) for all y ∈ D(A), where P is the Helmholtz projection operator from (L2(Ω))3 into X.
Time optimal control problems 15 / 21
The time optimal control problem now reads: (TP)M
3 :
T(M) inf
f ∈UM
To apply the above-mentioned abstract result to obtain the bang-bang property for (TP)M
3 , we write X = L2 σ(Ω) and U = {0} × L2(ω) × L2(ω).
Define the operator A on X by
0(Ω)
3 L2
σ(Ω),
Ay = P(∆y) for all y ∈ D(A), where P is the Helmholtz projection operator from (L2(Ω))3 into X. Let B ∈ L(U, X) be defined by Bf = Pf for all f ∈ U.
Time optimal control problems 15 / 21
The time optimal control problem now reads: (TP)M
3 :
T(M) inf
f ∈UM
To apply the above-mentioned abstract result to obtain the bang-bang property for (TP)M
3 , we write X = L2 σ(Ω) and U = {0} × L2(ω) × L2(ω).
Define the operator A on X by
0(Ω)
3 L2
σ(Ω),
Ay = P(∆y) for all y ∈ D(A), where P is the Helmholtz projection operator from (L2(Ω))3 into X. Let B ∈ L(U, X) be defined by Bf = Pf for all f ∈ U. Then, A is self-adjoint and generates an analytic semigroup in X;
Time optimal control problems 15 / 21
The time optimal control problem now reads: (TP)M
3 :
T(M) inf
f ∈UM
To apply the above-mentioned abstract result to obtain the bang-bang property for (TP)M
3 , we write X = L2 σ(Ω) and U = {0} × L2(ω) × L2(ω).
Define the operator A on X by
0(Ω)
3 L2
σ(Ω),
Ay = P(∆y) for all y ∈ D(A), where P is the Helmholtz projection operator from (L2(Ω))3 into X. Let B ∈ L(U, X) be defined by Bf = Pf for all f ∈ U. Then, A is self-adjoint and generates an analytic semigroup in X; B is an admissible control
B∗ϕ = (0, χωϕ2, χωϕ3) for all ϕ = (ϕ1, ϕ2, ϕ3) ∈ X;
Time optimal control problems 15 / 21
The time optimal control problem now reads: (TP)M
3 :
T(M) inf
f ∈UM
To apply the above-mentioned abstract result to obtain the bang-bang property for (TP)M
3 , we write X = L2 σ(Ω) and U = {0} × L2(ω) × L2(ω).
Define the operator A on X by
0(Ω)
3 L2
σ(Ω),
Ay = P(∆y) for all y ∈ D(A), where P is the Helmholtz projection operator from (L2(Ω))3 into X. Let B ∈ L(U, X) be defined by Bf = Pf for all f ∈ U. Then, A is self-adjoint and generates an analytic semigroup in X; B is an admissible control
B∗ϕ = (0, χωϕ2, χωϕ3) for all ϕ = (ϕ1, ϕ2, ϕ3) ∈ X; The equation (4) can be rewritten as dy dt = Ay + Bf , t > 0. y(0) = y0.
Time optimal control problems 15 / 21
On the other hand, by Coron-Guerrero’s work
Stokes system with N − 1 scalar controls, Journal of Differential Equations, 246 (2009), 2908-2921. there exists a positive constant C = C(Ω, ω) such that for each L ∈ (0, 1],
3
|ϕj(x, L)|2 dx ≤ e
C L9
L
|ϕ2(x, t)|2 + |ϕ3(x, t)|2 dxdt for all ϕ0 ∈ L2
σ(Ω), where ϕ = (ϕ1, ϕ2, ϕ3) solves the equation
ϕt − ∆ϕ + ∇p = 0, in Ω × (0, L), div ϕ = 0, in Ω × (0, L), ϕ = 0, in ∂Ω × (0, L), ϕ(·, 0) = ϕ0.
Time optimal control problems 16 / 21
In other words, the pair (A∗, B∗) satisfies observability inequality: eLA∗ϕ02
X ≤ e
C L9
L B∗etA∗ϕ02
U dt for all ϕ0 ∈ X and L ∈ (0, 1].
Time optimal control problems 17 / 21
In other words, the pair (A∗, B∗) satisfies observability inequality: eLA∗ϕ02
X ≤ e
C L9
L B∗etA∗ϕ02
U dt for all ϕ0 ∈ X and L ∈ (0, 1].
Therefore, we have
Corollary
Problem (TP)M
3 has the bang-bang property.
Time optimal control problems 17 / 21
In other words, the pair (A∗, B∗) satisfies observability inequality: eLA∗ϕ02
X ≤ e
C L9
L B∗etA∗ϕ02
U dt for all ϕ0 ∈ X and L ∈ (0, 1].
Therefore, we have
Corollary
Problem (TP)M
3 has the bang-bang property.
set is replaced by the pointwise-type:
the corresponding bang-bang property is much more difficult. This work is now in process co-worked with Felipe and Diego.
Time optimal control problems 17 / 21
1
Motivations
2
Main results
3
Conclusion and further research
Time optimal control problems 18 / 21
Different kinds of observability inequality from measurable subsets imply different versions of bang-bang property for time optimal control problems, as well as norm optimal control problems.
Time optimal control problems 18 / 21
Different kinds of observability inequality from measurable subsets imply different versions of bang-bang property for time optimal control problems, as well as norm optimal control problems. When the controlled system is not ”time-invariant”, we still do not know how to derive the bang-bang property even if we have established the corresponding observability inequality from measurable subsets.
Time optimal control problems 18 / 21
For the progress of B-B-P, we refer the following recent work, which in particular derive, but from another aspect, the bang-bang property of time
a(x, t) = a(x) + b(t) ∈ L∞.
bang-bang property of time optimal controls for heat equations. SIAM
Time optimal control problems 19 / 21
Study the bang-bang property of optimal control problems to the general parabolic equation with some nonlinearity terms. On this issue, the bang-bang property for the semilinear heat equation with global Lipschitz nonlinearity and a good sign condition is proved. For the general case, it is still open.
control of semilinear heat equation. Annales de I’Institut Henri Poincar´ e (C) Non Linear Analysis, 31 (2014), 477-499.
semilinear parabolic equation. Discrete Contin. Dyn. Syst. 36 (2016), 279-302.
Time optimal control problems 20 / 21
Time optimal control problems 21 / 21