On the solvability of a system of forward-backward linear equations with unbounded operator coefficients

On the solvability of a system of forward-backward linear
equations with unbounded operator coeﬃcients
Nikita V. Artamonov
MGIMO University, Moscow, Russia
5th Najman Conference, September 14, 2017
Nikita Artamonov (MGIMO) Solvability of FBEE September 14, 2017 1 / 32

Content
1 Introduction. Motivation
2 Triple of Banach spaces, operators, C0-semigroups
3 System of Forward-Backward Evolution Equations
4 Diﬀerential Operator Riccati Equation
5 Solvability of The System of FBEE

Introduction
Some control problems lead to the system of forward-backward evolution
equations
x (t) = −Ax(t) − By(t)
y (t) = A∗y(t) − Cx(t)
t ∈ [0, T]
with boundary condition
x(0) = x0 y(T) = Gx(T)
1st evolution equation is forward in time
2nd evolution equation is backward in time

1st Example: LQ-control
Consider LQ-control problem
min
u(·)∈U
1
2
T
0
(Qx(s), x(s)) + (Ru(s), u(s))ds +
1
2
(Gx(T), x(T))
s.t.
x (t) = −Ax(t) + Bu(t)
x(0) = x0
Pontryagin’s Minimal Principle leads to the system



x (t) = −Ax(t) − BR−1B∗y(t)
y (t) = A∗y(t) − Qx(t)
x(0) = x0 y(T) = Gx(T)

2nd Example: Mean Field Game
Introduced by Lasry-Lions and Huang-Caines-Malham´e since 2006
Main goal
Describe dynamics with large numbers (a continuum) of agents whose
strategies depend on the distribution law.
Features of the model:
players act according to the same principles
Their strategies takes into account the distribution of co-players.
Idea
Macroscopic description through a mean ﬁeld limit as the number of
players N → +∞.

Mean Field Game: agent strategy
Average agent solves control problem1
inf
βt
J(β) = E
T
0
[L(Xs, βs, m(s))]ds + G(XT , m(T))
s.t. dXt = βtdt +
√
2dWt
Here m(t) is the distribution law of Xt.
1
L may depend of t

Mean Field Game: Model
Mean Field Game System2: no (0, T) × Ω
mt = ∆m + div(mHp(x, m, u))
ut = −∆u + H(x, m, u)
with conditions (initial distribution and ﬁnal pay-oﬀ)
m(0) = m0 u(T) = G(x, m(T))
1st is Kolmogorov-Fokker-Plank eq. (forward in time)
2nd is Hamilton-Jacobi-Bellman eq. (backward in time)
2
non-linear in the general case

MFG: applications
Numbers of applications:
Production of limited resource
Propagation of behavior in the social areas (Mexican waves, fashions
etc)
Eﬀects of competition on the dynamics of human capital (distribution
of salaries etc)
Financial and risk management

Content

The triple of spaces
Consider a triple of
X± are Banach spaces
H is a Hilbert space
with continuous injective dense embeddings
X+ →
j+
H →
j−
X− (1)
Then
j = j−j+ : X+ → X−
is continuous injective dense embedding as well.
Notations: (·, ·) is the inner product in H.

The triple of spaces
We will assume that
X− = (X+)∗ with duality y, x (y ∈ X−, x ∈ X+)
X+ is reﬂexive
the duality is consistent with the Hilbert iner product
j−z, x = (z, j+x) ∀z ∈ H, x ∈ X+
Corollary
j− = (j+)∗
jy, x = (j+y, j+x) for all x, y ∈ X+
j = j∗

Operators in the triple of spaces
Let A ∈ L(X+, X−)
Consider the related operator pencil
Lλ = A + λj ∈ L(X+, X−) λ ∈ C (2)
Deﬁnition
The resolvent set of the pencil is deﬁned as
ρ(L) = {λ ∈ C | ∃L−1
λ ∈ L(X−, X+)}
The spectrum of the pencil is σ(L) = Cρ(L)

For given bounded operator A we introduce (unbounded) operators
Â = Aj−1
: dom(Â) ⊂ X− → X−
ˇA = j−1
A : dom(ˇA) ⊂ X+ → X+
¯A = j−1
− Aj−1
+ : dom(¯A) ⊂ H → H
with the domains
dom(¯A) = {x ∈ H|∃z ∈ X+ : x = j+z, Az ∈ j−H},
dom(ˇA) = {x ∈ X+|Ax ∈ jX+}.
dom(Â) = jX+
Remark
By the definition y = ¯Ax iff ∃z ∈ X+ such that x = j+z and Az = j−y

Proposition
Let ρ(L) = ∅. Then the operators ¯A, Â, ˇA are densely defined and closed.
More over,
1 ρ(L) ⊆ ρ(−Â), ρ(−ˇA) и ρ(L) = ρ(−¯A)
2 for all λ ∈ ρ(L) we have
dom(ˇA) = L−1
λ dom(Â) Lλ
ˇAL−1
λ = Â

Accretive operators in the triple of spaces
Deﬁnition
An operator A ∈ L(X+, X−) is called accretive, if
Re Ax, x ≥ 0 ∀x ∈ X+
If A is accretive and ρ(L) = ∅ than ¯A is m-accretive in the Hilbert space H
and
{λ ∈ C| Re λ > 0} ⊆ ρ(L) = ρ(−¯A) ⊆ ρ(−ˆA), ρ(−ˇA).

Semigroups in the triple of spaces
Theorem
Let A ∈ L(X+, X−) be accretive and ρ(L) = ∅. Then there exist uniformly
bounded C0-semigroups T−
t ∈ L(X−), U(t) ∈ L(H), T+
t ∈ L(X+) s.t.
1 −¯A is the generator of Ut in H, −ˆA is the generator of T−
t in X−, −ˇA
is the generator of T+
t in X+;
2 The following diagrams are commutative
X+
j+
−−−−→ H
j−
−−−−→ X−
T+
t


Ut


T−
t


X+
j+
−−−−→ H
j−
−−−−→ X−
X+
A
−−−−→ X−


T+
t


T−
t
X+
A
−−−−→ X−
3 for all x ∈ X+ in the space X−
(jT+
t x) = −AT+
t x = −T−
t Ax.

Adjoint operator in the triple of spaces
Since X− = (X+)∗ and X+ is reﬂexive, then
A ∈ L(X+, X−) =⇒ A∗
∈ L(X+, X−)
and ρ(A∗ + λj) = ρ(A + λj).
Obviously, if A is accretive then A∗ is accretive as well3.
For the adjoint operator we can introduce (unbounded) operators
¯A∗ = j−1
− A∗
j−1
+
ˆA∗ = A∗
j−1 ˇA∗ = j−1
A∗
in spaces H, X−, X+ respectively.
3
since A∗
x, y = Ay, x ∀x, y ∈ X+

Adjoint semigroups in the triple of spaces
Corollary
Under the assumptions of the Theorem 1 there exist semigroups
S+
t ∈ L(X+), Vt ∈ L(H), S−
t ∈ L(X−) (t ≥ 0) s.t.
1 −ˆA∗ is the generator of S−
t in X−, −¯A∗ is the generator of Vt in H
and −ˇA∗ is the generator of S+
t in X+.
2 j+S+
t = Vtj+, j−Vt = S−
t j− and A∗S+
t = S−
t A∗.
3 (jS+
t x) = −A∗S+
t x для всех x ∈ X+.
Proposition
Under the assumptions of the Theorem 1 and the Corollary 2
S+
t = (T−
t )∗
S−
t = (T+
t )∗

Content

Assumptions on operators
Consider operators A, B, C, G in the triple of spaces (1)
We shall assume
1 A ∈ L(X+, X−) is accretive and ρ(L) = ∅.
2 C, G ∈ L(X+, X−) are self-adjoint non-negative, i.e.
Cx, x , Gx, x ≥ 0 ∀x ∈ X+.
3 B ∈ L(X−, X+) is self-adjoint non-negative, i.e.
y, By ≥ 0 ∀y ∈ X−

System of FBEE
Consider a system of forward-backward linear evolution equations (FBEE)
on [0, T]
x (t) = −ˇAx(t) − By(t)
y (t) = ˆA∗y(t) − Cx(t)
x(t) ∈ X+
y(t) ∈ X−
(FBEE)
with boundary conditions
x(0) = x0 y(T) = Gx(T) (3)
1st equation is forward in time in X+
2nd equation is backward in time in X−.

System of FBEE: mild solution
Deﬁnition
Vector-functions x(t) ∈ X+ and y(t) ∈ X− (t ∈ [0, T]) are called mild
solution of the system (FBEE), if
x(t) = T+
t x(0) −
t
0
T+
s By(s)ds
y(t) = S−
T−ty(T) +
T
t
S−
s−tCx(s)ds
for all t ∈ [0, T].
Remark
First equation is equivalent
(jx(t)) = −Ax(t) − jBy(t)

Content

Related Riccati equation
Consider the solution of the system (FBEE) under the form
y(t) = P(t)x(t) P(t) ∈ L(X+, X−)
Formally substituting into the system (FBEE) we obtain operator
diﬀerential Riccati equation4
P (t) = −C + ˆA∗P(t) + P(t)ˇA + P(t)BP(t) (4)
with initial condition
P(T) = G (5)
4
backward in time

Riccati equation in the triplet of spaces
Deﬁnition
Continuous operator-function P(t) ∈ L(X+, X−) is called mild solution of
(backward in time) Riccati equation (4) with the initial condition (5) if it
satisﬁes the integral equation
P(t) = S−
T−tGT+
T−t +
T
t
S−
s−t C − P(s)BP(s) T+
s−tds
Here S−
t , T+
t are C0-semigroups in spaces X−, X+ generated by the
operators A∗, A respectively.

Solvability of Riccati equation
Theorem
Let the conditions (1), (2), (3) on the operators A, B, C, G are fulﬁlled.
Then the diﬀerentiala Riccati equation (4) with the initial condition (5) has
a unique mild solution P(t) = P∗(t) ≥ 0 ∈ L(X+, X−)
a
backward in time

Some related result: I. Lasiecka & S. Triggiani,
G. Da Prato & A. Ichikawa
Mild solution P(t) = P∗(t) ≥ 0 ∈ L(H) of Riccati equation
P (t) = −A∗
P(t) − P(t)A − C + P(t)BP(t) P(T) = G
Here A is sectorial in H and G, B, C ∈ L(H) are self-adjoint and
non-negative.

Some related result: A.J. Pritchard & D. Salamon
Triplet of Hilbert spaces H+ → H → H−
Let −A is a generator of a semigroup in H and exp(−tA) can be extened
to L(H−).
Mild solution P(t) = P∗(t) ≥ 0 ∈ L(H−, H+) of Riccati equation
P (t) = −A∗
P(t) − P(t)A − C + P(t)BP(t) P(T) = G
provided G, C ∈ L(H−, H+), B ∈ L(H+, H−) are non-negative
self-adjoint.

Content

Solvability of FBEE
Theorem
Let the conditions (1), (2), (3) on the operators A, B, C, G are fulﬁlled.
Then for all x0 ∈ X+ the system of forward-backward evolution equations
(FBEE) with boundary conditions (3) has a unique mild solution
x(t) ∈ X+ y(t) ∈ X− t ∈ [0, T].
in the sense of the Deﬁnition 3.

Solvability of FBEE
J. Young ’15: solvability5 of FBEE system
x (t) = −Ax(t) − By(t)
y(t) = A∗y(t) − Cx(t)
x(t), y(t) ∈ H
provided
1 G, B, C ∈ L(H) are self-adjoint and non-negative
2 −A is a generator of C0-semigroup in H.
5
in weak sense

Thank you for your attention

On the solvability of a system of forward-backward linear equations with unbounded operator coefficients

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