Deterministic Infinite-Dimensional Control Theory

• Deterministic infinite-dimensional optimal control theory investigates optimization for systems defined over Banach and Hilbert spaces, capturing complex dynamics with infinite degrees of freedom.
• Key methods include dynamic programming with HJB equations and variational formulations, which enable the derivation of feedback laws and necessary optimality conditions.
• Alternative formulations via occupation measures and linear programming provide dual representations, supporting computational approximations and deeper structural insights.

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 [(Fabbri et al., 24 Sep 2025); (Kipka et al., 2014); (Kamoutsi et al., 2017); (Borkar et al., 2018)]. 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 (Fabbri et al., 24 Sep 2025).

Other formulations are explicitly geometric. For control systems on infinite-dimensional manifolds, the state manifold $M$ is an infinite-dimensional $X$0-Banach manifold modeled on a reflexive Banach space $X$1 admitting a $X$2-smooth bump function with locally Lipschitz second derivative. The dynamics are

$X$3

with measurable controls $X$4 taking values in a complete separable metric space, and trajectories given by absolutely continuous maps into $X$5 (Kipka et al., 2014). 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 $X$6-term problem, one works with a Gelfand triple

$X$7

control space $X$8 a real Hilbert space, admissible controls in $X$9, and states in

$U$0

The state equation is

$U$1

for a twice continuously Fréchet-differentiable mapping $U$2 (Court et al., 2016).

Linear systems theory supplies a further canonical class. In the passive-systems formulation, the state space $U$3, input space $U$4, and output space $U$5 are separable Hilbert spaces. The system may be represented either as a well-posed linear system $U$6 in the sense of Tucsnak–Weiss or as a system node

$U$7

in the sense of Staffans and Salamon, allowing unbounded input and output operators (Hastir et al., 4 Jun 2025).

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: $U$8 and $U$9 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 (Berthier et al., 2021). 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

$A:D(A)\subset X\to X$0

and under admissibility axioms satisfies the Dynamic Programming Principle

$A:D(A)\subset X\to X$1

Formally, this yields the infinite-dimensional HJB equation

$A:D(A)\subset X\to X$2

with Hamiltonian

$A:D(A)\subset X\to X$3

and terminal condition $A:D(A)\subset X\to X$4 (Fabbri et al., 24 Sep 2025). Under compactness of the control set and sufficient regularity, one proves that $A:D(A)\subset X\to X$5 solves HJB pointwise; when these hypotheses fail, weaker notions such as viscosity or strong solutions are used (Fabbri et al., 24 Sep 2025).

The same HJB logic appears in weak and dual formulations. For smooth finite-horizon problems, the max-subsolution dual problem is

$A:D(A)\subset X\to X$6

subject to

$A:D(A)\subset X\to X$7

and

$A:D(A)\subset X\to X$8

for all $A:D(A)\subset X\to X$9 (Berthier et al., 2021). Under mild convexity and smoothness, the supremum equals $C_0$0.

For ensemble density control, the value object is itself a functional of a probability density. If $C_0$1 solves the Chapman–Kolmogorov partial integro-differential equation

$C_0$2

the value functional

$C_0$3

satisfies the functional HJB equation

$C_0$4

with terminal condition $C_0$5. The adjoint variable satisfies

$C_0$6

linking dynamic programming to the infinite-dimensional minimum principle (Bakshi et al., 2020).

The LP duals of long-run average problems are HJB-type inequalities rather than equalities. In continuous time,

$C_0$7

for all $C_0$8 (Borkar et al., 2018), while in discrete time the dual inequality takes the form

$C_0$9

for all $y'(s)=Ay(s)+F(s,y(s),u(s)),\qquad y(t)=x,$0 (Borkar et al., 2018). 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 (Fabbri et al., 24 Sep 2025).

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:

$y'(s)=Ay(s)+F(s,y(s),u(s)),\qquad y(t)=x,$1

The adjoint arc $y'(s)=Ay(s)+F(s,y(s),u(s)),\qquad y(t)=x,$2 satisfies

$y'(s)=Ay(s)+F(s,y(s),u(s)),\qquad y(t)=x,$3

almost everywhere, together with the transversality condition

$y'(s)=Ay(s)+F(s,y(s),u(s)),\qquad y(t)=x,$4

and the maximum condition

$y'(s)=Ay(s)+F(s,y(s),u(s)),\qquad y(t)=x,$5

almost everywhere (Kipka et al., 2014). The proof combines nonsmooth analysis, Lagrangian charts, chattering, and relaxed-control approximation. The abnormal multiplier case $y'(s)=Ay(s)+F(s,y(s),u(s)),\qquad y(t)=x,$6 may occur only when the endpoint constraint fails a certain weak controllability property toward the target set (Kipka et al., 2014).

For PDE-type systems with a free time parameter and an $y'(s)=Ay(s)+F(s,y(s),u(s)),\qquad y(t)=x,$7-type contribution to the cost, a change of variables fixes the horizon on $y'(s)=Ay(s)+F(s,y(s),u(s)),\qquad y(t)=x,$8 through a map $y'(s)=Ay(s)+F(s,y(s),u(s)),\qquad y(t)=x,$9 satisfying $u(\cdot)\in L^1_{\mathrm{loc}}([t,T];U)$0. The reformulated state equation is

$u(\cdot)\in L^1_{\mathrm{loc}}([t,T];U)$1

and the adjoint satisfies on $u(\cdot)\in L^1_{\mathrm{loc}}([t,T];U)$2 and $u(\cdot)\in L^1_{\mathrm{loc}}([t,T];U)$3

$u(\cdot)\in L^1_{\mathrm{loc}}([t,T];U)$4

with terminal and jump conditions

$u(\cdot)\in L^1_{\mathrm{loc}}([t,T];U)$5

Stationarity holds both in the control and in the free time:

$u(\cdot)\in L^1_{\mathrm{loc}}([t,T];U)$6

supplemented by KKT conditions for the control-energy constraint (Court et al., 2016). The same work derives second-order conditions on a critical cone and supports a Newton method based on Hessian-vector products (Court et al., 2016).

For density control of marked jump diffusions, the infinite-dimensional minimum principle introduces an adjoint function $u(\cdot)\in L^1_{\mathrm{loc}}([t,T];U)$7 and the Hamiltonian functional

$u(\cdot)\in L^1_{\mathrm{loc}}([t,T];U)$8

The costate equation is

$u(\cdot)\in L^1_{\mathrm{loc}}([t,T];U)$9

and stationarity requires $$y(s)=e^{(s-t)A}x+\int_t^s e^{(s-r)A}F(r,y(r),u(r))\,dr.$$0 (Bakshi et al., 2020). 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 $$y(s)=e^{(s-t)A}x+\int_t^s e^{(s-r)A}F(r,y(r),u(r))\,dr.$$1, the discounted occupation measure

$$y(s)=e^{(s-t)A}x+\int_t^s e^{(s-r)A}F(r,y(r),u(r))\,dr.$$2

satisfies the Liouville constraint

$$y(s)=e^{(s-t)A}x+\int_t^s e^{(s-r)A}F(r,y(r),u(r))\,dr.$$3

for every $$y(s)=e^{(s-t)A}x+\int_t^s e^{(s-r)A}F(r,y(r),u(r))\,dr.$$4, and the primal LP minimizes $$y(s)=e^{(s-t)A}x+\int_t^s e^{(s-r)A}F(r,y(r),u(r))\,dr.$$5 subject to $$y(s)=e^{(s-t)A}x+\int_t^s e^{(s-r)A}F(r,y(r),u(r))\,dr.$$6 (Kamoutsi et al., 2017). Its dual maximizes $$y(s)=e^{(s-t)A}x+\int_t^s e^{(s-r)A}F(r,y(r),u(r))\,dr.$$7 over smooth subsolutions of the HJB inequality

$$y(s)=e^{(s-t)A}x+\int_t^s e^{(s-r)A}F(r,y(r),u(r))\,dr.$$8

In discrete time, Gaitsgory–Parkinson–Shvartsman formulate both discounted and average criteria through occupational measures on the viability graph

$$y(s)=e^{(s-t)A}x+\int_t^s e^{(s-r)A}F(r,y(r),u(r))\,dr.$$9

For the long-run average problem, the feasible set is

$B:U\to X_{-1}$0

and the primal value is $B:U\to X_{-1}$1, while the dual is

$B:U\to X_{-1}$2

The discounted analogue uses a modified balance equation containing $B:U\to X_{-1}$3 and yields $B:U\to X_{-1}$4 exactly (Gaitsgory et al., 2017).

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

$B:U\to X_{-1}$5

the occupational-measure accumulation points lie in

$B:U\to X_{-1}$6

and a more precise formulation introduces an additional nonnegative measure to encode the initial condition (Borkar et al., 2018). The continuous-time analogue likewise augments the primal by a second non-negative measure $B:U\to X_{-1}$7 through

$B:U\to X_{-1}$8

for all $B:U\to X_{-1}$9 (Borkar et al., 2018).

Problem class	Primal balance equation	Dual inequality
Discounted continuous time	$M$0	$M$1
Long-run average discrete time	$M$2	$M$3
Non-ergodic average cost	Stationarity plus an extra nonnegative measure encoding the initial condition	$M$4

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 (Kamoutsi et al., 2017, Borkar et al., 2018). In the non-ergodic discrete-time case, if $M$5 is continuous, then $M$6, and along an optimal trajectory one has

$M$7

with the converse optimality implication also valid (Borkar et al., 2018). Another structural result is the asymptotic passage from discounted to average formulations:

$M$8

in discrete time (Gaitsgory et al., 2017).

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 (Borkar et al., 2018, Borkar et al., 2018).

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 $M$9, $X$00, and compact basic semi-algebraic $X$01 satisfying Putinar’s condition, Lasserre’s hierarchy produces primal moment and dual sum-of-squares relaxations. The truncated moment sequence $X$02 defines a moment matrix $X$03 and localizing matrices $X$04, and the SDP relaxation satisfies

$X$05

Combined with the equivalence theorem, this yields $X$06 (Kamoutsi et al., 2017). The same framework also supports extraction of an approximate feedback controller through

$X$07

The RKHS-based “kernel-SoS” approach extends sum-of-squares reasoning beyond polynomial data. With $X$08 and an RKHS $X$09 of functions on $X$10, Hamiltonian nonnegativity is strengthened to the existence of a positive semidefinite operator $X$11 such that

$X$12

After subsampling, this becomes a finite SDP in $X$13 (Berthier et al., 2021). The representation is exact in special cases: infinite-horizon LQR and smooth control-affine systems with $X$14 of class $X$15 and strongly convex (Berthier et al., 2021). Under RKHS sampling bounds, the SDP solution converges to the true value, and if the Hamiltonian lies in a Sobolev RKHS of smoothness $X$16, the error decays like

$X$17

For control problems with a free switching or maximizing time, the numerical strategy in the $X$18-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 (Court et al., 2016). 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,

$X$19

which leads to a sampling-based backward sweep, a control gradient

$X$20

and the open-loop update

$X$21

Because each grid-point update and each trajectory simulation is independent, the algorithm “parallelizes trivially” (Bakshi et al., 2020).

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

$X$22

passivity implies finite cost for every $X$23, and there exists a unique bounded, self-adjoint, nonnegative operator $X$24 such that

$X$25

Thus the optimal-cost operator is a contraction (Hastir et al., 4 Jun 2025). In the impedance energy-preserving case

$X$26

one has the explicit solution

$X$27

and an adapted operator Riccati equation reduces to the familiar gain form when $X$28 and $X$29 (Hastir et al., 4 Jun 2025). Applications include boundary control systems, first-order port-Hamiltonian systems, and an Euler–Bernoulli beam with shear-force control (Hastir et al., 4 Jun 2025).

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 (Fabbri et al., 24 Sep 2025). 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 (Fabbri et al., 24 Sep 2025).

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 (Kipka et al., 2014). 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 (Kamoutsi et al., 2017). Classical HJB solutions may fail to exist in important infinite-dimensional economic models, requiring viscosity or strong-solution methods instead (Fabbri et al., 24 Sep 2025). Normality of Pontryagin extremals may fail when weak controllability toward endpoint constraints is absent (Kipka et al., 2014). 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 (Hastir et al., 4 Jun 2025).

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.