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Econ-126 · economics · Economics

Uniform turnpike estimates toward time-varying reference trajectories

LLM status Open unverified

Statement

SC-10 |  Stochastic control and dynamic optimization

Research agenda formalization

Definitions and assumptions

Let $X\subset\R^n$ and $U\subset\R^r$ be compact constraint sets. Consider a family of continuous-time problems \[ \min\int_0^T\ell(t,x(t),u(t))dt,\qquad \dot x=f(t,x,u),\quad x(0)=x_0,\quad x(T)\in X_f. \] Controls are measurable, states absolutely continuous, and the constraints hold almost everywhere. Assume $f,\ell$ are continuous, $f$ is uniformly Lipschitz in $x$, and optimal pairs exist for the horizons and endpoints under consideration. For a concrete subclass, take $f,\ell$ periodic in $t$ with period $P>0$, and fix an admissible $P$-periodic reference pair $(\bar x,\bar u)$ minimizing average cost among such periodic pairs. Initial states and terminal constraints range over a specified compact, uniformly feasible family.

Formal question

Find verifiable assumptions under which every optimal pair $(x_T,u_T)$ has, for each $\varepsilon>0$, a constant $C_\varepsilon$ independent of $T$ and the allowed endpoints such that \[ \operatorname{Leb}\{t\in[0,T]: |x_T(t)-\bar x(t)|+|u_T(t)-\bar u(t)|>\varepsilon\} \leq C_\varepsilon. \] Determine which strengthened conditions imply exponential estimates $|x_T-\bar x|+|u_T-\bar u| \leq C(e^{-\lambda t}+e^{-\lambda(T-t)})$ almost everywhere, with $C,\lambda>0$ independent of $T$.

Scope and status

Periodicity and average-cost optimality alone do not imply these bounds. This formalization asks for a continuous-time theory that specifies the reference trajectory, dissipativity or stability hypotheses, and uniform endpoint reachability. The cited survey identifies the direction; it does not establish that every later periodic special case remains open.

References for the source of the problem

Timm Faulwasser and Lars Grüne, Turnpike Properties in Optimal Control: An Overview of Discrete-Time and Continuous-Time Results, arXiv:2011.13670v3 (16 March 2021), §5, "Topics not Discussed and Open Problems," p. 25, paragraph on time-varying and periodic turnpikes. See §2 for the turnpike notions. https://arxiv.org/pdf/2011.13670v3.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-126-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-126

Uniform periodic turnpikes from dissipation and scalar tracking

Catalogue problem: C61-123. Status: Unresolved. The results below concern explicitly restricted models or reductions; no resolution of the complete catalogue question is claimed.

Abstract

A bounded time-dependent storage function and uniformly bounded entrance and exit costs imply a measure turnpike toward a specified periodic reference. For a scalar integrator with periodic quadratic tracking cost, a maximum-principle argument gives explicit horizon-independent exponential estimates for both state and control. Endpoints and compact constraints are treated explicitly. The general constrained nonlinear theory remains outside these results.

Question and reference

For compact state and control constraint sets, consider \[ \min\int_0^T\ell(t,x(t),u(t))\,dt, \quad \dot x=f(t,x,u),\quad x(0)=x_0,\quad x(T)\in X_f. \] A prescribed admissible periodic pair $(\bar x(t),\bar u(t))$ minimizes periodic average cost. The question asks when every optimal pair spends a bounded amount of time away from that same-phase reference, uniformly in $T$ and allowed endpoints, and when stronger exponential bounds hold. Faulwasser and Grüne discuss this direction and distinguish time-varying data from autonomous systems with periodic optimal motions [FG].

Put $\bar\ell(t)=\ell(t,\bar x(t),\bar u(t))$ and $d(t,x,u)=|x-\bar x(t)|+|u-\bar u(t)|$. A time shift of the reference is not silently allowed.

A measure estimate with explicit uniform assumptions

Assume there are constants $K_S,M_\ell,\tau>0$, a continuously differentiable storage function $S(t,x)$, and a continuous increasing function $\alpha:[0,\infty)\to[0,\infty)$ with $\alpha(0)=0$, $\alpha(r)>0$ for $r>0$, such that:
  1. $|S(t,x)|\le K_S$ on all feasible states and times, and $|\ell(t,x,u)-\bar\ell(t)|\le M_\ell$ on feasible triples.
  2. Every feasible pair satisfies, almost everywhere, \[ \ell(t,x,u)-\bar\ell(t)\ge \alpha(d(t,x,u))+\partial_tS(t,x)+\nabla_xS(t,x)\cdot f(t,x,u). \tag{1} \]
  3. For every allowed endpoint problem with $T\ge2\tau$, there is a feasible entrance arc on $[0,\tau]$ from $x_0$ to $\bar x(\tau)$ and a feasible exit arc on $[T-\tau,T]$ from $\bar x(T-\tau)$ to $X_f$. The reference is feasible between them.
The exit assumption is uniform in the terminal phase $T\bmod P$. It cannot be replaced by controllability at just one phase.

Theorem (Uniform measure turnpike): Under these assumptions, every optimal pair satisfies, for every $\varepsilon>0$, \[ \operatorname{Leb}\{t\in[0,T]:d(t,x_T,u_T)>\varepsilon\} \le \max\left\{2\tau, \frac{2\tau M_\ell+2K_S}{\alpha(\varepsilon)}\right\}. \tag{2} \] The bound is independent of the allowed endpoints and of the horizon.
Proof: For $T\ge2\tau$, concatenate the entrance arc, the same-phase reference, and the exit arc. Its excess cost relative to $\int_0^T\bar\ell(t)dt$ is at most $2\tau M_\ell$. Optimality gives the same upper bound for the optimal excess cost. Integrating (1) along the optimal trajectory and using the ordinary chain rule for an absolutely continuous state gives \[ \int_0^T\alpha(d(t,x_T,u_T))dt \le 2\tau M_\ell-S(T,x_T(T))+S(0,x_0) \le2\tau M_\ell+2K_S. \] On the set where $d>\varepsilon$, monotonicity implies $\alpha(d)\ge\alpha(\varepsilon)>0$. Dividing yields (2). For $T<2\tau$ the measure is at most $T<2\tau$, proving the remaining case. ∎

The theorem is a direct dissipativity argument with its required constants made explicit. It does not claim that a storage function satisfying (1) always exists for a periodic average-cost minimizer.

An exponential result for nonconstant periodic tracking

Let $\bar x$ be a $C^2$ periodic scalar function and $\bar u=\dot{\bar x}$. Let $q(t)$ be continuous and periodic with $0<q_0\le q(t)\le q_1$, and fix $r>0$. For a horizon $T\ge T_0>0$ and endpoint errors $|a|,|b|\le B$, consider \[ \dot x=u,\quad x(0)=\bar x(0)+a,\quad x(T)=\bar x(T)+b, \qquad \ell(t,x,u)=\tfrac12q(t)(x-\bar x(t))^2+ \tfrac12r(u-\bar u(t))^2. \tag{3} \] The reference has zero cost and is the unique zero-cost periodic pair. Define \[ \kappa=\sqrt{q_0/r},\qquad d_0=\min\{1,T_0/2\},\qquad C_u=e^{\kappa d_0}\left(\frac2{d_0}+\frac{q_1d_0}{2r}\right). \tag{4} \] First allow unconstrained scalar state and control, and afterwards choose compact constraints containing the bounds below.
Theorem (Explicit exponential state and control estimates): Problem (3) has a unique optimal pair, with $x$ twice continuously differentiable. For every $T\ge T_0$ and every allowed endpoint pair, \[ |x(t)-\bar x(t)|\le B\{e^{-\kappa t}+e^{-\kappa(T-t)}\}, \tag{5} \] \[ |u(t)-\bar u(t)|\le BC_u\{e^{-\kappa t}+e^{-\kappa(T-t)}\} \qquad (0\le t\le T). \tag{6} \] The same pair is the unique constrained optimum whenever the compact state and control intervals contain, respectively, \[ [\min\bar x-2B,\max\bar x+2B],\qquad [\min\bar u-2BC_u,\max\bar u+2BC_u]. \] For terminal sets contained in $\bar x(T)+[-B,B]$, every attained constrained optimum obeys (5)--(6), provided these interval conditions hold.
Proof: Set $z=x-\bar x$ and $v=u-\bar u$, so $\dot z=v$ and the objective is $\tfrac12\int_0^T(qz^2+r(z')^2)dt$. On the affine $H^1$ space with endpoints $a,b$, it is strictly convex and coercive; a minimizing sequence is bounded in $H^1$, has a weakly convergent subsequence with the same endpoint traces, and lower semicontinuity gives a unique minimizer. Its first variation against zero-endpoint smooth functions is zero, giving \[ -rz''+q(t)z=0. \tag{7} \] Continuity of $q$ implies $z\in C^2$ by integrating the weak equation.

Let $w(t)=B(e^{-\kappa t}+e^{-\kappa(T-t)})$. Its endpoint values dominate $|a|,|b|$, and $-rw''+qw=(q-q_0)w\ge0$. The scalar maximum principle gives $z\le w$ and $-z\le w$: if $z-w$ had a positive interior maximum, its second derivative would be nonpositive there, while $-r(z-w)''+q(z-w)\le0$ would be strictly positive, a contradiction. This proves (5).

At every $t\in[0,T]$ there is a one-sided interval of length $d_0$ contained in $[0,T]$, since $T\ge2d_0$. On a right interval, Taylor's integral identity yields \[ |z'(t)|\le\frac{|z(t+d_0)-z(t)|}{d_0} +\frac1{d_0}\int_t^{t+d_0}(t+d_0-s)|z''(s)|ds \le\left(\frac2{d_0}+\frac{q_1d_0}{2r}\right) \sup_{|s-t|\le d_0}|z(s)|. \] The left-interval identity gives the same bound. Estimate (5) on this interval is at most $Be^{\kappa d_0}(e^{-\kappa t}+e^{-\kappa(T-t)})$, proving (6).

The interval conditions make the unconstrained optimizer feasible for the compactly constrained problem, so it remains optimal; strict convexity gives uniqueness. Finally fix the terminal point actually chosen by an optimum with a terminal set. The just-constructed optimizer for that fixed terminal point is feasible and uniquely minimizes among all paths with those endpoints. The given optimum must coincide with it, and its endpoint error is at most $B$, proving the last assertion. ∎

The lower horizon bound is needed for a uniform control constant: if $a\ne b$ is fixed and $T\downarrow0$, the identity $\int_0^Tz'(t)dt=b-a$ forces $\|z'\|_\infty\ge|b-a|/T$. Thus a theorem covering arbitrarily short horizons needs a correspondingly restricted endpoint family.

Why periodic optimality is insufficient

Even uniform ability to enter and leave a reference does not select it among multiple periodic minimizers. Take $\dot x=u$, $X=[-1,1]$, $U=[-1,1]$, and $\ell(x,u)=(1-x^2)^2+u^2$. Both $(x,u)=(1,0)$ and $(-1,0)$ are periodic average-cost minimizers with cost zero. Use reference $(1,0)$ and endpoints $x(0)=x(T)=-1$. The optimal path stays at $-1$ and is at distance two from the reference for the whole horizon. Entrance and exit bridges to the reference each exist in time two, with uniformly bounded cost, but a strict inequality such as (1) toward that particular reference cannot hold. The missing separation is therefore substantive.

The scalar exponential theorem deals with a uniformly positive tracking curvature and an inactive-constraint region. It does not prove analogous bounds for multidimensional nonlinear Hamiltonian systems, active state constraints, or nonunique periodic orbits. A general solution needs verifiable stability or dichotomy conditions and compatible endpoint manifolds; those are not supplied by periodicity alone. The results here are complete sufficient results in their declared subclasses, while the catalogue's broader continuous-time theory remains unresolved.

References

  • [FG] T. Faulwasser and L. Grüne (2021). Turnpike Properties in Optimal Control: An Overview of Discrete-Time and Continuous-Time Results. arXiv:2011.13670v3, Section 2 for notions, Section 5, p. 25 for time-varying and periodic directions. https://arxiv.org/pdf/2011.13670v3.

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