From ODE-Constrained Optimization to Bochner Spaces

Overview¶

The previous lectures developed the continuous and discrete theory for elliptic optimal control, including:

• reduced and all-at-once formulations;

• adjoint-based gradient representations;

• variational inequalities and normal-cone KKT systems;

• projected-gradient and primal-dual viewpoints.

At this stage, the next genuine change of perspective is the introduction of time-dependent dynamics. The simplest rigorous entry point is not yet a parabolic PDE, but a linear control problem governed by an ordinary differential equation. This lecture has two goals:

1. derive the optimality system for a linear-quadratic control problem governed by an ODE;

2. introduce the vector-valued function spaces needed to pass from ODEs to parabolic PDEs.

The logical path of the lecture is the following:

1. formulate a linear-quadratic control problem on $(0,T)$;

2. prove well-posedness of the state equation and continuity of the control-to-state map;

3. derive the reduced derivative through a linearized state equation;

4. introduce the adjoint ODE and obtain the gradient formula;

5. state the first-order optimality conditions, both unconstrained and box-constrained;

6. isolate the structural difference with the elliptic case: a forward-backward optimality system;

7. introduce strongly measurable vector-valued functions and Bochner spaces;

8. define weak time derivatives in dual spaces and the space

   $$W(0,T):=\{y\in L^2(0,T;V): y_t\in L^2(0,T;V')\};$$

   (1)

9. state the basic embedding and integration-by-parts results that make parabolic optimal control possible.

Model Problem¶

Let $T>0$ be fixed, and let

Let also

and let $\alpha>0$, $\beta\ge 0$. The control space is

The admissible set is a nonempty, closed, convex subset

For $u\in U$, the state $y=y(u)$ is defined by the linear ODE

We consider the optimal control problem

where

The three terms are:

• a distributed tracking term in time;

• the Tikhonov regularization on the control;

• an optional terminal observation.

This model already contains all structural ingredients of parabolic optimal control:

• an evolution equation;

• a cost integrated over time;

• a terminal contribution;

• an adjoint equation with a final condition.

State Equation and Control-to-State Map¶

We begin with the well-posedness of the state equation. Since the right-hand side belongs to $L^2(0,T;\mathbb R^n)$, one expects the solution to belong to $H^1(0,T;\mathbb R^n)$.

Define

Recall that in finite dimensions

hence the terminal value $y(T)$ is meaningful.

Proposition. For every $u\in U$, the state equation admits a unique solution

Moreover, the solution is given by the variation-of-constants formula

and there exists a constant $C>0$, depending only on $A$, $B$, and $T$, such that

Proof. Fix $u\in U$ and define

Since $A$ is a constant matrix, the matrix exponential $e^{tA}$ is well defined and continuous. The formula

defines a continuous function on $[0,T]$. Differentiating under the integral sign yields

for a.e. $t\in(0,T)$, and clearly $y(0)=y_0$. Thus $y\in H^1(0,T;\mathbb R^n)$ and solves the ODE.

Uniqueness follows from linearity. Indeed, if $y_1$ and $y_2$ solve the same problem, then $w:=y_1-y_2$ satisfies

Hence $w(t)=e^{tA}w(0)=0$ for all $t$.

To estimate $y$, let

Then

Taking the supremum in $t$ gives

Since $\dot y=Ay+g$,

Combining the bounds for $y$ and $\dot y$ yields the claim. $\square$

The proposition allows us to define the control-to-state map

Because the state equation is linear, $S$ is linear and continuous. The reduced functional is therefore

Existence and Uniqueness of an Optimal Control¶

The previous lectures already established an abstract existence theorem in Hilbert spaces. In the present setting the argument becomes particularly transparent.

Theorem. Assume $\alpha>0$ and let $U_{\mathrm{ad}}\subset U$ be nonempty, closed, and convex. Then the reduced problem

admits a unique solution $\bar u\in U_{\mathrm{ad}}$.

Proof. Since $S:U\to Y$ is continuous and $J$ is continuous on $Y\times U$, the map $j:U\to \mathbb R$ is continuous. Moreover,

Hence $j$ is coercive on $U$. Let $(u_k)$ be a minimizing sequence in $U_{\mathrm{ad}}$. Coercivity implies that $(u_k)$ is bounded in $U$. Since $U$ is a Hilbert space, there exists a subsequence, not relabeled, and an element $\bar u\in U$ such that

Because $U_{\mathrm{ad}}$ is closed and convex, it is weakly closed, hence $\bar u\in U_{\mathrm{ad}}$. By continuity and linearity of $S$,

Since the trace map

is a continuous linear functional on $H^1(0,T;\mathbb R^n)$, we also have

Since the norm is weakly lower semicontinuous in Hilbert spaces, each term in $j$ is weakly lower semicontinuous, hence

so $\bar u$ is a minimizer.

To prove uniqueness, let $u_1\neq u_2$ and $\theta\in(0,1)$. Since $S$ is linear,

The first and third terms of $j$ are convex, while the control term is strictly convex because $\alpha>0$:

Therefore $j$ is strictly convex and the minimizer is unique. $\square$

Linearized State Equation¶

To differentiate the reduced cost, we perturb the control by $h\in U$. Since the state equation is linear, the state increment is described by a linear ODE independent of the base point.

Let $u\in U$ and $h\in U$. Define

Then $z_h$ solves

Because the control-to-state map is linear, in fact

for every $u\in U$. Equivalently, one may derive the linearized equation directly from

by replacing $u$ with $u+\varepsilon h$, subtracting the equation for $u$, dividing by $\varepsilon$, and passing to the limit.

The directional derivative of the reduced functional therefore reads

The difficulty is that this formula still contains the linearized state $z_h$. As in the elliptic case, the adjoint variable removes this dependence.

Adjoint Equation¶

The adjoint equation is the backward-in-time equation dual to the linearized state equation. Its role is exactly the same as in the elliptic case, but the time direction is reversed.

Let $u\in U$ be fixed, and let $y=S(u)$. We define the adjoint state $p\in H^1(0,T;\mathbb R^n)$ by

This is again a linear ODE, now with terminal condition prescribed at $t=T$. Its unique solution is obtained by solving backward in time, or equivalently by the change of variable $s=T-t$.

Proposition. Let $u\in U$, $y=S(u)$, and let $p$ solve the adjoint equation above. Then for every $h\in U$, with $z_h$ solving the linearized state equation,

Proof. From the linearized state equation,

Multiply by $p$ and integrate over $(0,T)$:

Using $(p,Az_h)=(A^Tp,z_h)$ and integrating by parts in time,

Since $z_h(0)=0$,

By the adjoint equation,

hence

Finally, the terminal condition gives

which yields the claim. $\square$

Reduced Gradient Formula¶

We can now eliminate the linearized state from the derivative.

Theorem. The reduced functional $j:U\to \mathbb R$ is Fréchet differentiable and, for every $u\in U$,

where $p$ is the adjoint associated with $u$. Hence the gradient of $j$ in the Hilbert space $U=L^2(0,T;\mathbb R^m)$ is

Proof. We already computed

The adjoint identity from the previous proposition gives

Therefore

Since this is a bounded linear functional of $h$, the Fréchet derivative is represented in $U$ by the function $\alpha u + B^T p$. $\square$

First-Order Optimality System¶

We can now write the necessary and sufficient optimality conditions. Because the problem is convex and the reduced functional is strictly convex, first-order conditions characterize the unique minimizer.

Unconstrained case¶

If

then the stationarity condition is simply

i.e.

Thus the optimal control satisfies the explicit relation

The optimality system becomes

This is a forward-backward system:

• the state evolves forward from $t=0$;

• the adjoint evolves backward from $t=T$.

This two-sided time structure is the first major structural difference from elliptic control.

Constrained case¶

Assume now that $U_{\mathrm{ad}}\subset U$ is closed and convex. Then $\bar u\in U_{\mathrm{ad}}$ is optimal if and only if

Using the adjoint representation of the derivative, this becomes

Equivalently, in normal-cone form,

Hence the full optimality system is

Box Constraints and Projection Formula¶

A particularly important case is the box-constrained set

where $u_a,u_b\in L^\infty(0,T;\mathbb R^m)$ and the inequalities are understood componentwise.

In this case the optimality condition is equivalent to the pointwise projection formula

where for $\xi\in \mathbb R^m$

componentwise.

The proof is identical in structure to the elliptic case:

• the variational inequality is posed in the Hilbert space $L^2(0,T;\mathbb R^m)$;

• the admissible set is a closed convex box;

• the projection is characterized pointwise almost everywhere.

Thus the only genuinely new analytical ingredient introduced by time dependence is not the control condition, but the forward-backward evolution structure of state and adjoint.

Forward-Backward Interpretation¶

It is worth isolating the conceptual meaning of the adjoint in the dynamical case.

For a fixed control $u$:

• the state equation propagates the effect of the control from time 0 to time $T$;

• the adjoint equation propagates sensitivity information from time $T$ backward to time 0.

The terminal condition

encodes the derivative of the terminal observation. If $\beta=0$, then $p(T)=0$ and only the distributed tracking term drives the adjoint. If instead the cost is purely terminal,

then the adjoint satisfies

This is exactly the same phenomenon that one encounters later for parabolic PDEs:

• state equation forward in time;

• adjoint equation backward in time;

• control condition coupling state and adjoint at the same time level.

From ODEs to Evolution Equations¶

The ODE model can be rewritten abstractly as

where now the state $y(t)$ is an element of the finite-dimensional Hilbert space $H=\mathbb R^n$ and

For parabolic PDEs, the same formula remains formally correct, but the state at each time is no longer a vector in $\mathbb R^n$. Instead:

• $y(t)$ is a function of the space variable, e.g. $y(t)\in H_0^1(\Omega)$ or $L^2(\Omega)$;

• the derivative $y_t$ generally belongs to a dual space rather than to the same space as $y$;

• time integration must be carried out for vector-valued functions.

This is the reason why the usual scalar-valued Lebesgue and Sobolev spaces are not sufficient. We need function spaces of the form

where $X$ is itself a Banach or Hilbert space. These are the Bochner spaces.

Strongly Measurable Vector-Valued Functions¶

Let $X$ be a Banach space. A function

is called simple if it has the form

where $x_k\in X$ and $E_k\subset(0,T)$ are measurable sets.

A function $y:(0,T)\to X$ is called strongly measurable if there exists a sequence of simple functions $(y_n)$ such that

This is the natural notion of measurability for vector-valued functions. In the Hilbert spaces used in parabolic PDEs, separability holds, so this definition behaves well.

If $y$ is strongly measurable and

then $y$ is Bochner integrable and one may define

as the limit of the integrals of simple approximations. This is the vector-valued analogue of the usual Lebesgue integral.

Bochner Spaces $L^p(0,T;X)$¶

Let $1\le p<\infty$. We define

The norm is

Similarly,

Standard facts:

• if $X$ is Banach, then $L^p(0,T;X)$ is Banach;

• if $X$ is Hilbert and $p=2$, then $L^2(0,T;X)$ is Hilbert with inner product

  $$(y,z)_{L^2(0,T;X)} := \int_0^T (y(t),z(t))_X\,dt;$$

  (78)

• if $X\hookrightarrow Z$ continuously, then

  $$L^p(0,T;X)\hookrightarrow L^p(0,T;Z)$$

  (79)

  continuously.

Fundamental examples¶

Let $\Omega\subset \mathbb R^d$ be a bounded Lipschitz domain and set

Then:

1. if $X=L^2(\Omega)$,

   $$L^2(0,T;L^2(\Omega)) \cong L^2(Q_T);$$

   (81)

2. if $X=H_0^1(\Omega)$,

   $$L^2(0,T;H_0^1(\Omega))$$

   (82)

   consists of functions square integrable in time with values in $H_0^1(\Omega)$;

3. if $X=H^{-1}(\Omega)$,

   $$L^2(0,T;H^{-1}(\Omega))$$

   (83)

   is the natural space for weak time derivatives of parabolic states.

Thus, in parabolic theory, the same function $y(x,t)$ is viewed as a map

with values in a spatial function space.

Weak Time Derivatives¶

For parabolic equations, the time derivative does not usually belong to the same space as the state. This forces a dual-space formulation.

Let $X$ be a Banach space and let

A function

is called the weak time derivative of $y$ if for every scalar test function $\varphi\in C_c^\infty(0,T)$ and every $\ell\in X'$,

In this case we write

If $X$ is Hilbert, one can identify $X\cong X'$ by the Riesz map and recover the familiar scalar definition.

The Sobolev space of $X$-valued functions is then

For ODEs, where $X=\mathbb R^n$, this is the space used for the state and adjoint variables.

For parabolic PDEs, however, the correct setting is generally not $H^1(0,T;X)$ with a single space $X$, but a mixed space involving a Hilbert triple.

Gelfand Triples and the Space $W(0,T)$¶

Let $V$ and $H$ be Hilbert spaces such that

continuously and densely. By identifying $H$ with its dual $H'$ through the Riesz isomorphism, one obtains the Gelfand triple

The last embedding is defined by

The canonical parabolic example is

The natural energy space for parabolic problems is

It is a Hilbert space with norm

This is the parabolic analogue of $H^1(0,T;\mathbb R^n)$ for ODEs.

The key point is the asymmetry:

• the state lives in $V$ with respect to the space variable;

• the time derivative lives in the dual space $V'$.

This is forced by the weak formulation of the PDE. For the heat equation,

one expects

Fundamental Theorem for $W(0,T)$¶

The space $W(0,T)$ has a decisive property: its elements possess a continuous representative with values in $H$. This is what makes initial and terminal conditions meaningful.

Theorem (Lions-Magenes). Let

be a Gelfand triple. Then:

1. every $y\in W(0,T)$ admits a representative, still denoted by $y$, such that

   $$y\in C([0,T];H);$$

   (99)

2. the embedding

   $$W(0,T) \hookrightarrow C([0,T];H)$$

   (100)

   is continuous;

3. if $y,v\in W(0,T)$, then the scalar map

   $$t\mapsto (y(t),v(t))_H$$

   (101)

   is absolutely continuous and satisfies

   $$\frac{d}{dt}(y(t),v(t))_H = \langle y_t(t),v(t)\rangle_{V',V} + \langle v_t(t),y(t)\rangle_{V',V}$$

   (102)

   for a.e. $t\in(0,T)$.

In particular, taking $v=y$ gives

Integrating between $s$ and $t$ yields the energy identity

More generally, for $y,v\in W(0,T)$,

This is the time-integration-by-parts formula needed for parabolic adjoints.

Abstract Parabolic Problem¶

With the previous tools in place, one can formulate the prototype parabolic state equation.

Let $V\hookrightarrow H\hookrightarrow V'$ be a Gelfand triple. Let

be a bilinear form satisfying:

1. continuity:

   $$|a(w,v)|\le M\|w\|_V\|v\|_V \qquad \forall w,v\in V;$$

   (107)

2. coercivity:

   $$a(v,v)\ge c_a \|v\|_V^2 \qquad \forall v\in V,$$

   (108)

   for some $c_a>0$.

Define the operator

Given

the weak parabolic problem is: find $y\in W(0,T)$ such that

with

Theorem. Under the assumptions above, the parabolic problem admits a unique solution

Moreover,

for a constant $C$ depending only on the continuity and coercivity constants and on $T$.

This theorem is the infinite-dimensional analogue of the well-posedness proposition for the ODE state equation. The analogies are exact:

• matrix $A$ becomes an operator $A:V\to V'$;

• state space $H^1(0,T;\mathbb R^n)$ becomes $W(0,T)$;

• the terminal value is meaningful because $W(0,T)\hookrightarrow C([0,T];H)$.

Heat Equation as Canonical Example¶

Take

and define

Then

corresponds to the operator $A=-\Delta$ in weak form. Given a control $u\in L^2(0,T;L^2(\Omega))$, one may write

The parabolic state equation becomes

for all $v\in H_0^1(\Omega)$ and a.e. $t\in(0,T)$, with $y(0)=y_0$.

This is the standard weak formulation of the heat equation

with homogeneous Dirichlet boundary condition.

What Changes in the Optimality System for Parabolic PDEs?¶

At the formal level, almost nothing changes. The parabolic optimal control problem has the structure