---
title: Deterministic Infinite-Dimensional Control Theory
url: https://www.emergentmind.com/topics/deterministic-infinite-dimensional-optimal-control-theory
type: topic
---

# Deterministic Infinite-Dimensional Control Theory

Deterministic infinite-dimensional optimal control theory studies the optimization of deterministic dynamical systems whose natural description involves infinite-dimensional state spaces, infinite-dimensional manifolds, or equivalent formulations over infinite-dimensional spaces of measures, functions, and operators. In the cited literature, the subject encompasses abstract evolution equations on Hilbert spaces, control systems on infinite-dimensional $C^2$-Banach manifolds, occupation-measure formulations of discounted and long-run average problems, and Hamiltonian characterizations through dynamic programming, Hamilton–Jacobi–Bellman equations, and Pontryagin-type necessary conditions [2509.19909; 1405.3996; 1703.09005; 1812.04790]. Its central themes are viability, weak or mild solvability, duality between trajectory and measure formulations, and the recovery of optimal controls from adjoint or dual objects.

## 1. Functional-analytic settings and admissible dynamics

A standard abstract setting uses real separable Hilbert spaces $X$ and $U$, a possibly unbounded operator $A:D(A)\subset X\to X$ generating a $C_0$-semigroup, and a control system
$$
y'(s)=Ay(s)+F(s,y(s),u(s)),\qquad y(t)=x,
$$
with controls $u(\cdot)\in L^1_{\mathrm{loc}}([t,T];U)$ and mild solutions
$$
y(s)=e^{(s-t)A}x+\int_t^s e^{(s-r)A}F(r,y(r),u(r))\,dr.
$$
This is the basic semigroup framework emphasized in the economic survey of Fabbri–Faggian–Federico–Gozzi, including the extension to unbounded control operators $B:U\to X_{-1}$ through extrapolation spaces [2509.19909].

Other formulations are explicitly geometric. For control systems on infinite-dimensional manifolds, the state manifold $M$ is an infinite-dimensional $C^2$-Banach manifold modeled on a reflexive Banach space $E$ admitting a $C^2$-smooth bump function with locally Lipschitz second derivative. The dynamics are
$$
\dot q(t)=f(t,q(t),u(t)),\qquad q(0)=q_0,
$$
with measurable controls $u:[0,T]\to U$ taking values in a complete separable metric space, and trajectories given by absolutely continuous maps into $M$ [1405.3996]. This setting is designed for problems arising from partial differential equations with symmetry.

A variational framework appears in problems with PDE state equations and nonsmooth objectives. In the $L^\infty$-term problem, one works with a Gelfand triple
$$
Y\subset X\equiv X'\subset Y',
$$
control space $U$ a real Hilbert space, admissible controls in $L^2(0,T;U)$, and states in
$$
W(0,T;Y):=L^2(0,T;Y)\cap W^{1,2}(0,T;Y').
$$
The state equation is
$$
\dot y(t)=f(y(t),u(t))\ \text{in }Y',\qquad y(0)=y_0\in X,
$$
for a twice continuously Fréchet-differentiable mapping $f:Y\times U\to Y'$ [1608.08422].

Linear systems theory supplies a further canonical class. In the passive-systems formulation, the state space $X$, input space $U$, and output space $Y$ are separable Hilbert spaces. The system may be represented either as a well-posed linear system $\Sigma=(T,\Phi,\Psi,F)$ in the sense of Tucsnak–Weiss or as a system node
$$
S=\begin{pmatrix}A&B\\ C&D\end{pmatrix}
$$
in the sense of Staffans and Salamon, allowing unbounded input and output operators [2506.03882].

A recurrent misconception is that “infinite-dimensional” necessarily refers only to the state space. The RKHS-based sums-of-squares formulation makes the opposite point explicit: $X\subset\mathbb R^d$ and $U\subset\mathbb R^p$ may be finite-dimensional, while the admissible control space and the space of candidate value functions remain infinite-dimensional Banach or reproducing-kernel Hilbert spaces [2110.07396]. This suggests that deterministic infinite-dimensional optimal control theory is defined as much by its analytical objects as by the dimension of the physical state itself.

## 2. Dynamic programming and Hamilton–Jacobi–Bellman structures

The dynamic programming approach begins with the value function
$$
V(t,x)=\sup_{u\in\mathcal U(t,x)}J(t,x;u),
$$
and under admissibility axioms satisfies the Dynamic Programming Principle
$$
V(t,x)=\sup_{u\in\mathcal U(t,x)}\left[\int_t^r g(s,y(s),u(s))\,ds+V(r,y(r))\right],\qquad r\in(t,T].
$$
Formally, this yields the infinite-dimensional HJB equation
$$
- V_t(t,x)=\langle x,A^*D_xV(t,x)\rangle+\mathcal H(t,x,D_xV(t,x)),
$$
with Hamiltonian
$$
\mathcal H(t,x,p)=\sup_{u\in\mathcal C}\{\langle F(t,x,u),p\rangle+g(t,x,u)\},
$$
and terminal condition $V(T,x)=\phi(x)$ [2509.19909]. Under compactness of the control set and sufficient regularity, one proves that $V\in C^1$ solves HJB pointwise; when these hypotheses fail, weaker notions such as viscosity or strong solutions are used [2509.19909].

The same HJB logic appears in weak and dual formulations. For smooth finite-horizon problems, the max-subsolution dual problem is
$$
\sup_{V\in C^1([0,T]\times X)} \int_X V(0,x)\,d\mu_0(x)
$$
subject to
$$
H_V(t,x,u):=\frac{\partial V}{\partial t}(t,x)+L(t,x,u)+\nabla_xV(t,x)^\top f(t,x,u)\ge 0,
$$
and
$$
V(T,x)\le M(x),
$$
for all $(t,x,u)\in[0,T]\times X\times U$ [2110.07396]. Under mild convexity and smoothness, the supremum equals $\mathbb E_{x_0\sim\mu_0}[V^*(0,x_0)]$.

For ensemble density control, the value object is itself a functional of a probability density. If $\rho$ solves the Chapman–Kolmogorov partial integro-differential equation
$$
\partial_t\rho=\mathcal A_t[u]\rho,
$$
the value functional
$$
V(t,\rho)=\inf_{u(\cdot)}\left\{\int_t^T\langle \ell(s,\cdot,u(s)),\rho(s)\rangle\,ds+\langle \phi,\rho(T)\rangle\right\}
$$
satisfies the functional HJB equation
$$
-\partial_tV(t,\rho)=\inf_u H\bigl(\rho,u,\delta V/\delta\rho\bigr),
$$
with terminal condition $V(T,\rho)=\langle \phi,\rho\rangle$. The adjoint variable satisfies
$$
\lambda(t,x)\approx \delta V/\delta\rho(t,\rho(t))(x),
$$
linking dynamic programming to the infinite-dimensional minimum principle [2009.07154].

The LP duals of long-run average problems are HJB-type inequalities rather than equalities. In continuous time,
$$
\ell(x,u)-\nabla v(x)\cdot f(x,u)\ge \lambda
$$
for all $(x,u)\in X\times U$ [1805.02311], while in discrete time the dual inequality takes the form
$$
g(y,u)+h(f(y,u))-h(y)\ge \lambda
$$
for all $(y,u)\in Y\times U^0$ [1812.04790]. These inequalities encode lower bounds on the optimal value and become equalities along optimal trajectories under no-duality-gap hypotheses.

An important limitation is that classical HJB theory is not universally available in infinite dimensions. The economic models highlighted by Fabbri–Faggian–Federico–Gozzi include state constraints, non-Lipschitz data, and non-regularizing differential operators, precisely the cases in which standard semigroup smoothing arguments may fail [2509.19909].

## 3. Pontryagin principles, adjoint equations, and first-order optimality

For control systems on infinite-dimensional manifolds, the Pontryagin Maximum Principle furnishes first-order necessary conditions for optimality in pure Mayer form:
$$
\min \ \ell(q(0),q(T)) \quad \text{subject to } (q(0),q(T))\in S.
$$
The adjoint arc $p(\cdot)\in AC([0,T],T^*M)$ satisfies
$$
-\dot p(t)\in \partial_q H(t,q^*(t),p(t),u^*(t))
$$
almost everywhere, together with the transversality condition
$$
(p(0),-p(T))\in \partial_L\ell(q^*(0),q^*(T))+N_S(q^*(0),q^*(T)),
$$
and the maximum condition
$$
H(t,q^*(t),p(t),u^*(t))=\sup_{v\in U}H(t,q^*(t),p(t),v)
$$
almost everywhere [1405.3996]. The proof combines nonsmooth analysis, Lagrangian charts, chattering, and relaxed-control approximation. The abnormal multiplier case $A_0=0$ may occur only when the endpoint constraint fails a certain weak controllability property toward the target set [1405.3996].

For PDE-type systems with a free time parameter and an $L^\infty$-type contribution to the cost, a change of variables fixes the horizon on $[0,2]$ through a map $\pi(\cdot,\tau)$ satisfying $\pi(1,\tau)=\tau$. The reformulated state equation is
$$
\dot y(s)=\pi'(s,\tau)f(y(s),u(s)),\qquad y(0)=y_0,
$$
and the adjoint satisfies on $(0,1)$ and $(1,2)$
$$
-\dot p(s)=\pi'(s,\tau)H_y(y(s),u(s),p(s),\lambda),
$$
with terminal and jump conditions
$$
p(2)=D\phi_2(y(2)),\qquad p(1^+)-p(1^-)+D\phi_1(y(1))=0.
$$
Stationarity holds both in the control and in the free time:
$$
0=\pi'(s,\tau)H_u(y(s),u(s),p(s),\lambda),\qquad
0=\int_0^2 \pi'_\tau(s)H(y(s),u(s),p(s),\lambda)\,ds,
$$
supplemented by KKT conditions for the control-energy constraint [1608.08422]. The same work derives second-order conditions on a critical cone and supports a Newton method based on Hessian-vector products [1608.08422].

For density control of marked jump diffusions, the infinite-dimensional minimum principle introduces an adjoint function $\lambda(t,x)$ and the Hamiltonian functional
$$
H(\rho(t,\cdot),u(t),\lambda(t,\cdot))
=
\langle \ell(t,\cdot,u(t)),\rho(t,\cdot)\rangle
+\langle \lambda(t,\cdot),\mathcal A_t[u]\rho(t,\cdot)\rangle.
$$
The costate equation is
$$
-\partial_t\lambda(t,x)=\ell(t,x,u(t))+\mathcal A_t[u]^\dagger \lambda(t,x),
\qquad \lambda(T,x)=\phi(x),
$$
and stationarity requires $\delta H/\delta u=0$ [2009.07154]. This is the infinite-dimensional analogue of the classical minimum principle, but the state and adjoint now evolve as a forward density and a backward PIDE.

A useful synthesis is that adjoint methods persist across very different infinite-dimensional settings, but the adjoint object changes with the ambient geometry: covector arcs on Banach manifolds, dual-space trajectories in variational PDE settings, or backward functionals over densities.

## 4. Occupation measures and infinite-dimensional linear programming

The LP approach replaces trajectories by occupation measures and recasts deterministic control as an infinite-dimensional linear program over measures. In continuous time with discount factor $\lambda>0$, the discounted occupation measure
$$
\mu(E)=\int_0^\infty e^{-\lambda t}\delta_{(x(t),u(t))}(E)\,dt
$$
satisfies the Liouville constraint
$$
\int_{X\times U}\bigl[\lambda\phi(x)-\nabla\phi(x)\cdot f(x,u)\bigr]\,d\mu(x,u)=\phi(x_0),
$$
for every $\phi\in C^1(X)$, and the primal LP minimizes $\int g\,d\mu$ subject to $A^*\mu=\delta_{x_0}$ [1703.09005]. Its dual maximizes $\phi(x_0)$ over smooth subsolutions of the HJB inequality
$$
\lambda\phi(x)-\nabla\phi(x)\cdot f(x,u)\le g(x,u).
$$

In discrete time, Gaitsgory–Parkinson–Shvartsman formulate both discounted and average criteria through occupational measures on the viability graph
$$
G=\{(y,u)\mid y\in Y,\ u\in U(y),\ f(y,u)\in Y\}.
$$
For the long-run average problem, the feasible set is
$$
W=\left\{\mu\in P(G)\ \middle|\ \int[\phi(f(y,u))-\phi(y)]\,d\mu(dy,du)=0,\ \forall \phi\in C(Y)\right\},
$$
and the primal value is $\inf_{\mu\in W}\int g\,d\mu=g^*$, while the dual is
$$
\sup_{\phi\in LS(Y)}\inf_{(y,u)\in G}\{g(y,u)+\phi(f(y,u))-\phi(y)\}=g^*.
$$
The discounted analogue uses a modified balance equation containing $(1-\alpha)[\phi(y_0)-\phi(y)]$ and yields $(1-\alpha)V_\alpha(y_0)$ exactly [1702.00857].

Borkar–Gaitsgory–Shvartsman extend the average-cost LP theory to the non-ergodic case, where the long-run value may depend on the initial condition. In the discrete-time system
$$
y(t+1)=f(y(t),u(t)),\qquad y(0)=y_0,
$$
the occupational-measure accumulation points lie in
$$
W=\left\{\gamma\in\mathcal P(Y\times U^0)\ \middle|\ \int[\phi(y)-\phi(f(y,u))]\,d\gamma(y,u)=0,\ \forall\phi\in C(Y)\right\},
$$
and a more precise formulation introduces an additional nonnegative measure to encode the initial condition [1812.04790]. The continuous-time analogue likewise augments the primal by a second non-negative measure $\nu$ through
$$
\int \nabla\varphi\cdot f\,d\mu+\int(\varphi(x)-\varphi(x_0))\,d\nu=0,
$$
for all $\varphi\in C^1(X)$ [1805.02311].

| Problem class | Primal balance equation | Dual inequality |
|---|---|---|
| Discounted continuous time | $\int[\lambda\phi-\nabla\phi\cdot f]\,d\mu=\phi(x_0)$ | $\lambda\phi-\nabla\phi\cdot f\le g$ |
| Long-run average discrete time | $\int[\phi(f(y,u))-\phi(y)]\,d\mu=0$ | $g(y,u)+\phi(f(y,u))-\phi(y)\ge g^*$ |
| Non-ergodic average cost | Stationarity plus an extra nonnegative measure encoding the initial condition | $g(y,u)+h(f(y,u))-h(y)\ge \lambda$ |

These LP formulations are not merely relaxations. Under compactness, continuity, viability, and no-duality-gap assumptions, the primal optimum, the dual optimum, and the optimal control value coincide [1703.09005; 1812.04790]. In the non-ergodic discrete-time case, if $V(\cdot)=\lim V_T(\cdot)/T$ is continuous, then $V(y_0)=d^*(y_0)$, and along an optimal trajectory one has
$$
g(y(t),u(t))+h(f(y(t),u(t)))-h(y(t))=V(y_0),\qquad h(y(t))=h(y_0),
$$
with the converse optimality implication also valid [1812.04790]. Another structural result is the asymptotic passage from discounted to average formulations:
$$
\lim_{\alpha\to1^-}\min_{y_0\in Y}(1-\alpha)V_\alpha(y_0)=g^*,
\qquad
\lim_{S\to\infty}\min_{y_0\in Y}\frac1S V(S,y_0)=g^*
$$
in discrete time [1702.00857].

A common misconception is that long-run average LP theory is inherently ergodic. The non-ergodic formulations show instead that initial-condition dependence can be retained explicitly by augmenting the primal measure constraints [1812.04790; 1805.02311].

## 5. Structured subclasses and computational approximations

When the data are polynomial, the occupation-measure LP admits semidefinite approximations. For deterministic continuous-time infinite-horizon discounted control with polynomial $f$, $g$, and compact basic semi-algebraic $X\times U$ satisfying Putinar’s condition, Lasserre’s hierarchy produces primal moment and dual sum-of-squares relaxations. The truncated moment sequence $z$ defines a moment matrix $M_r(z)$ and localizing matrices $M_{r-v_i}(q_i z)$, and the SDP relaxation satisfies
$$
J_r^*(x_0)=J_r(x_0),\qquad J_r(x_0)\uparrow J(x_0)\quad \text{as } r\to\infty.
$$
Combined with the equivalence theorem, this yields $J_r\to V_x(x_0)$ [1703.09005]. The same framework also supports extraction of an approximate feedback controller through
$$
u_r^*(x)=\arg\min_{u\in U}\bigl[g(x,u)-(\lambda\phi_r^*(x)-\nabla\phi_r^*(x)\cdot f(x,u))\bigr].
$$

The RKHS-based “kernel-SoS” approach extends sum-of-squares reasoning beyond polynomial data. With $E=[0,T]\times X\times U$ and an RKHS $\mathcal H$ of functions on $E$, Hamiltonian nonnegativity is strengthened to the existence of a positive semidefinite operator $A\in\mathcal S_+(\mathcal H)$ such that
$$
H_V(y)=\langle \Phi(y),A\Phi(y)\rangle_{\mathcal H},\qquad y\in E.
$$
After subsampling, this becomes a finite SDP in $(\theta,B)$ [2110.07396]. The representation is exact in special cases: infinite-horizon LQR and smooth control-affine systems with $u\mapsto L(t,x,u)$ of class $C^s$ and strongly convex [2110.07396]. Under RKHS sampling bounds, the SDP solution converges to the true value, and if the Hamiltonian lies in a Sobolev RKHS of smoothness $s$, the error decays like
$$
O\!\bigl(n^{-\tfrac{s}{d+p+1}}\bigr).
$$

For control problems with a free switching or maximizing time, the numerical strategy in the $L^\infty$-term setting is “optimize-then-discretize”: Crank–Nicolson time discretization, finite elements in PDE examples, Barzilai–Borwein gradient steps with Armijo line-search, followed by Newton’s method on the KKT system solved by GMRES [1608.08422]. The analysis is tightly coupled to the derived first- and second-order optimality conditions.

For density control, the backward PIDE for the costate is avoided through a linear Feynman–Kac representation,
$$
\lambda(t,x)=E_{X_t=x}^{u(\cdot)}\left[\phi(X_T)+\int_t^T \ell(s,X_s,u(s))\,ds\right],
$$
which leads to a sampling-based backward sweep, a control gradient
$$
g(t_i)=\delta H/\delta u\big|_{\rho,u,\lambda},
$$
and the open-loop update
$$
u^{k+1}(t_i)=u^k(t_i)-\alpha g^k(t_i).
$$
Because each grid-point update and each trajectory simulation is independent, the algorithm “parallelizes trivially” [2009.07154].

A plausible implication is that modern computation in deterministic infinite-dimensional control is increasingly organized around dual certificates: Riccati operators, HJB subsolutions, RKHS SoS operators, or sampled adjoint fields.

## 6. Canonical applications, recurring difficulties, and current perspectives

A central solvable subclass is infinite-horizon LQ control for passive systems. For well-posed linear systems or system nodes with
$$
J(x_0,u)=\int_0^\infty \|y(t)\|_Y^2+\|u(t)\|_U^2\,dt,
$$
passivity implies finite cost for every $x_0$, and there exists a unique bounded, self-adjoint, nonnegative operator $\Pi$ such that
$$
V(x_0)=\langle \Pi x_0,x_0\rangle_X,\qquad 0\le \Pi\le I_X.
$$
Thus the optimal-cost operator is a contraction [2506.03882]. In the impedance energy-preserving case
$$
A=-A^*,\qquad B=C^*,\qquad D=0,
$$
one has the explicit solution
$$
\Pi=I_X,\qquad u^{\mathrm opt}(t)=-\,y(t)=-\,C x(t),
$$
and an adapted operator Riccati equation reduces to the familiar gain form when $D=0$ and $R=I$ [2506.03882]. Applications include boundary control systems, first-order port-Hamiltonian systems, and an Euler–Bernoulli beam with shear-force control [2506.03882].

Applied economic modeling supplies a different class of applications. The survey by Fabbri–Faggian–Federico–Gozzi emphasizes spatial AK-type growth, transboundary pollution, vintage-capital models with delay, vintage-capital PDEs with boundary control, and time-to-build models [2509.19909]. In these examples, Dynamic Programming is often preferred because explicit or quasi-explicit solutions of the associated HJB equations can be obtained. The examples also display recurring analytical obstacles: state constraints, positivity constraints, non-regularizing semigroups, and the failure of the standard regularity theory used in finite-dimensional control [2509.19909].

The manifold PMP framework identifies another significant application domain: optimal control problems for partial differential equations invariant under a Lie-group action, where the state manifold is a manifold of functions such as Hilbert or Sobolev spaces [1405.3996]. Here the role of geometry is not incidental; it is embedded in the state representation and in the definition of admissible variations.

Several points remain structurally delicate. Strong duality in LP formulations is not automatic and depends on compactness, lower-semicontinuity, and no-relaxation-gap assumptions [1703.09005]. Classical HJB solutions may fail to exist in important infinite-dimensional economic models, requiring viscosity or strong-solution methods instead [2509.19909]. Normality of Pontryagin extremals may fail when weak controllability toward endpoint constraints is absent [1405.3996]. Even in highly structured linear-quadratic settings, uniqueness of the stabilizing optimal control may depend on additional properties such as approximate controllability or strong stability of the closed-loop semigroup [2506.03882].

Taken together, these developments show a field organized around several complementary representations of optimality. Dynamic programming characterizes value functions and feedback laws; Pontryagin principles encode first-order necessary conditions through adjoint variables; occupation-measure LPs linearize infinite-horizon criteria on spaces of measures; and specialized subclasses such as passive LQ systems admit explicit Riccati-type solutions. The unifying feature is that deterministic control problems, once lifted to the appropriate infinite-dimensional analytic setting, can often be reformulated so that optimality is expressed as equality or duality between trajectories, measures, functions, and operators.

Source: https://www.emergentmind.com/topics/deterministic-infinite-dimensional-optimal-control-theory