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Transversality condition

In the calculus of variations, the transversality condition is a necessary optimality criterion applied to problems with variable endpoints, where the terminal point of the extremal curve lies on a prescribed manifold rather than being fixed. It stipulates that the variation of the functional must vanish at the boundary, typically requiring the extremal to intersect the terminal curve orthogonally or satisfy a specific differential relation derived from the Euler-Lagrange equations.^[1] This condition ensures that small perturbations at the endpoint do not alter the functional's value to first order, thereby confirming the extremal's minimality or stationarity.^[2] For problems minimizing an integral functional J = \int_{x_1}^{x_2} F(x, y, y') \, dx with one fixed endpoint (x_1, y_1) and the other variable on a curve \phi(x_2, y_2) = 0, the transversality condition takes the form F \, dx_2 + (dy_2 - y' \, dx_2) F_{y'} = 0 at the terminal point.^[1] This can simplify to perpendicularity for geometric problems, such as the shortest path from a point to a line, where the extremal (a straight line) meets the line at a right angle.^[2] When both endpoints vary, the condition applies symmetrically at each, often leading to natural boundary conditions like F_{y'} = 0 if the terminal manifold is vertical.^[1] In optimal control theory, transversality conditions extend these ideas to dynamic systems, appearing as boundary constraints in Pontryagin's maximum principle for problems with free terminal time t_f or state x(t_f). For instance, in a system minimizing J = \int_{t_0}^{t_f} g(x, u, t) \, dt subject to \dot{x} = f(x, u, t) and a terminal constraint m(x(t_f), t_f) = 0, the condition requires the costate \lambda(t_f) = \nu \nabla_x m(t_f) and the Hamiltonian H(t_f) + \nu \frac{\partial m}{\partial t}(t_f) = 0, where H(t) = \lambda(t) \cdot f(x(t), u(t), t) - g(x(t), u(t), t) and \nu is a multiplier.^[3] This prevents "information loss" at the horizon and is crucial for infinite-horizon problems, where it often manifests as a discounted limit condition like \lim_{t \to \infty} e^{-\rho t} \lambda(t) x(t) = 0 to rule out non-optimal paths.^[4] These conditions, first systematically developed in the early 20th century—formally introduced around 1900 by mathematicians such as M. Kneser—alongside Euler-Lagrange theory, underpin applications in physics (e.g., brachistochrone problems with free endpoints), economics (dynamic optimization of growth models), and engineering (trajectory optimization in aerospace).^[1] Extensions to fractional calculus and higher-order derivatives preserve the core idea, adapting the condition to non-integer order integrals for more general variational problems.^[5]

Introduction

Definition

In the calculus of variations, the transversality condition arises in problems where the endpoint of the extremal curve is not fixed but constrained to lie on a specified terminal curve, serving as a necessary condition for the curve to extremize the functional J = \int_{t_0}^{t_1} L(t, y(t), y'(t)) \, dt.^[6] Suppose the terminal point (t_1, y(t_1)) lies on a curve parameterized as y = \phi(t); then, for an admissible variation, the first variation \delta J = 0 requires that the boundary term at t_1 vanishes, yielding the transversality condition

\left[ L + L_{y'} (\phi'(t_1) - y'(t_1)) \right]_{t=t_1} = 0,

where L_{y'} = \frac{\partial L}{\partial y'}.^[6] This equation ensures that the optimal path intersects the terminal curve such that the direction of the extremal is aligned with the geometry of the boundary, specifically making the variation orthogonal to the constraint.^[6] Geometrically, the condition implies that at the endpoint, the optimal curve touches the terminal manifold in a way that its tangent is transversal to the manifold's tangent space, preventing any first-order change in the functional value along admissible perturbations confined to the boundary.^[1] This transversality guarantees the minimality (or maximality) of the functional among nearby curves satisfying the endpoint constraint.^[1] In optimal control theory, an analogous condition applies to the terminal values of the costate variables, enforcing similar boundary requirements for the adjoint system.^[2]

Historical Development

The transversality condition originated in the development of the calculus of variations during the 18th century, where early variational problems with free or variable endpoints necessitated boundary constraints beyond fixed conditions. Leonhard Euler laid foundational groundwork in his 1744 treatise Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, implicitly addressing free-endpoint scenarios in problems like the brachistochrone through necessary conditions for extrema, marking the inception of variational methods that later formalized transversality. Euler explicitly derived conditions for free endpoints in problems like the catenary with variable attachment points in his 1766 work, providing early instances of transversality requirements.^[7] Joseph-Louis Lagrange advanced this framework in his 1788 Mécanique Analytique, introducing multiplier techniques to handle constraints in variational integrals, which provided essential tools for incorporating boundary variations and influenced subsequent derivations of endpoint conditions.^[8] In the 19th century, Karl Weierstrass elevated these ideas to a rigorous level in his 1879 lectures on the calculus of variations, where he developed necessary conditions for extrema, including the excess function for strong minima, and provided a general treatment of problems with variable boundaries, addressing limitations in earlier geometric approaches.^[9] David Hilbert's work around 1900, including his proof of the Dirichlet principle, further shaped the calculus of variations by rigorously justifying variational methods, countering Weierstrass's earlier critiques on existence, and contributing to boundary value problems through direct methods. Gilbert Ames Bliss synthesized and expanded these developments in his 1925 textbook Calculus of Variations, offering a comprehensive treatment of transversality for problems with movable endpoints and accessory boundary conditions, solidifying its place in classical variational theory.^[10] The mid-20th century saw the transversality condition integrated into optimal control theory, notably through Lev Pontryagin's maximum principle developed in the 1950s amid Soviet efforts in space trajectory optimization, where it appeared as terminal boundary requirements for Hamiltonian systems to ensure optimality in dynamic problems.^[11] In economics, Paul Samuelson applied variational techniques, including transversality, to dynamic optimization in his 1947 Foundations of Economic Analysis, particularly in chapters on intertemporal equilibrium and capital accumulation, bridging classical variations to economic growth models.^[12] Kenneth Arrow extended these applications in the 1950s through works on optimal investment and resource allocation, such as his collaborations on dynamic programming, where transversality conditions enforced sustainable paths in infinite-horizon economic models.^[13]

Calculus of Variations

Variable Endpoint Formulation

In the variable endpoint formulation of calculus of variations, the problem involves minimizing a functional of the form

J = \int_{a}^{b} F(x, y, y') \, dx,

where the initial point is fixed at y(a) = y_a, but the terminal point (b, y(b)) is constrained to lie on a specified curve y = \phi(x), with b free.^[14] This setup contrasts with fixed endpoint problems, where both boundary values are prescribed, and requires additional conditions to ensure optimality at the variable boundary.^[1] The integrand F is assumed to be sufficiently smooth, typically twice differentiable with respect to its arguments, to allow for the necessary variational analysis.^[9] To derive the necessary conditions, consider admissible variations \delta y(x) that perturb the extremal curve y(x) while preserving the initial condition \delta y(a) = 0 and keeping the terminal point on the curve \phi. For a variation parameterized by a small \epsilon, the perturbed endpoint satisfies y(b + \delta b) + \delta y(b + \delta b) = \phi(b + \delta b), leading to the constraint \delta y(b) = \phi'(b) \delta b + o(\delta b).^[14] The first variation of the functional, obtained via integration by parts and the Leibniz rule for varying limits, yields boundary terms at x = b: F(b) \delta b + F_{y'}(b) [\delta y(b) - y'(b) \delta b]. Substituting the endpoint constraint gives the integrated first variation as zero for an extremal, implying the transversality condition

F(b) + F_{y'}(b) [\phi'(b) - y'(b)] = 0

at the terminal point.^[1] This condition ensures that the variation in J vanishes to first order for all admissible perturbations.^[9] Specific cases arise depending on the orientation of the terminal curve. When the terminal manifold is vertical—corresponding to a vertical line where δx = 0 and δy is free—the transversality condition simplifies to the natural boundary condition F_{y'}(b) = 0.^[14] For a general slanted boundary, the condition involves both F and F_{y'}, relating the slope of the extremal y'(b) to that of the terminal curve \phi'(b). In the case of a horizontal terminal manifold (fixed y(b), free b), it becomes F(b) - y'(b) F_{y'}(b) = 0.^[1] A representative example is the shortest path from a fixed point to a straight line, where the functional is the arc length J = \int_{a}^{b} \sqrt{1 + (y')^2} \, dx. Here, F = \sqrt{1 + (y')^2} and F_{y'} = y' / \sqrt{1 + (y')^2}, so the transversality condition implies $1 + y'(b) \phi'(b) = 0, or y'(b) = -1 / \phi'(b). For a horizontal line (\phi' = 0), this yields a vertical intersection, but in general, the optimal straight-line geodesic meets the line at right angles, perpendicular to its tangent.^[9] This geometric interpretation underscores the condition's role in ensuring the extremal "reflects" optimally at the boundary.^[1]

Derivation and Geometric Interpretation

In the calculus of variations, the transversality condition emerges as a necessary requirement for the first variation of the functional to vanish in problems where the terminal endpoint lies on a prescribed manifold. Consider the functional J = \int_{a}^{b} F(x, y, y') \, dx, where the initial point is fixed at (a, y(a)) and the terminal point (b, y(b)) varies along a curve in the (x, y)-plane, such as y = \psi(x). To derive the condition, compute the first variation \delta J under admissible variations \delta y = h(x) and endpoint perturbations \delta x and \delta y(b) that remain tangent to the terminal manifold.^[6]^[15] The first variation is given by

\delta J = \int_{a}^{b} \left( F_y h + F_{y'} h' \right) dx + \left[ F_{y'} h \right]_{a}^{b},

where the subscripts denote partial derivatives and the boundary term at x = a vanishes due to the fixed initial point. Integrating the second term by parts yields

\int_{a}^{b} F_{y'} h' \, dx = \left[ F_{y'} h \right]_{a}^{b} - \int_{a}^{b} \frac{d}{dx} (F_{y'}) h \, dx,

\delta J = \int_{a}^{b} \left( F_y - \frac{d}{dx} F_{y'} \right) h \, dx + F_{y'}(b) h(b).

Accounting for the variable endpoint, the full boundary contribution at b becomes F_{y'} \delta y + (F - y' F_{y'}) \delta x, where \delta y and \delta x satisfy the tangency condition to the terminal curve, such as \delta y = \psi'(b) \delta x. For \delta J = 0 at an extremal, the Euler-Lagrange equation F_y - \frac{d}{dx} F_{y'} = 0 holds in the interior, and the boundary term must satisfy

\left[ F_{y'} \delta y + (F - y' F_{y'}) \delta x \right]_{b} = 0

for all such tangent variations. Substituting the relation between \delta y and \delta x gives the transversality condition

F_{y'} \psi' + (F - y' F_{y'}) = 0 \quad \text{at} \quad (b, y(b)).

^[6]^[15]^[9] Geometrically, this condition implies that at the terminal point, the vector (F_{y'}, -(F - y' F_{y'})) is normal to the tangent vector (dy, -dx) of the terminal curve in the (y, x)-plane, ensuring the optimal extremal intersects the manifold orthogonally with respect to this duality. In the standard (x, y)-plane, for geometric variational problems such as minimizing arc length, the extremal curve meets the terminal boundary such that its direction is perpendicular to the boundary curve; in general, the intersection satisfies the duality condition derived from the first variation, analogous to the path of light rays in Fermat's principle of least time, where reflection occurs at the interface to minimize optical path length.^[6]^[15] This can be visualized, in geometric problems, as an extremal arc approaching a curved terminal line in the plane, striking it at right angles: the slope of the extremal y' satisfies the condition such that the angle of incidence equals the angle of reflection if the manifold represents a reflective boundary.^[6]

Optimal Control Theory

Finite-Horizon Boundary Conditions

In finite-horizon optimal control problems, the transversality condition specifies the terminal values of the costate variables to ensure optimality at the end of the fixed time interval [t_0, T]. The standard problem involves minimizing the objective functional

J = \int_{t_0}^{T} L(t, x(t), u(t))\, dt + g(x(T))

subject to the state dynamics \dot{x}(t) = f(t, x(t), u(t)), with fixed initial conditions x(t_0) = x_0 and free terminal state x(T). This setup captures a wide range of engineering and economic applications where the horizon ends at a predetermined time T, and no constraints are imposed on the final state. The Pontryagin maximum principle provides the necessary conditions for optimality, centered on the Hamiltonian function H(t, x, u, \lambda) = L(t, x, u) + \lambda^\top f(t, x, u), where \lambda(t) \in \mathbb{R}^n is the costate vector associated with the state x(t). The costate evolves according to \dot{\lambda}(t) = -\frac{\partial H}{\partial x}(t, x(t), u(t), \lambda(t)), and the optimal control u^*(t) maximizes H pointwise. The transversality condition links the terminal costate to the objective: for a scalar terminal cost g, it requires \lambda(T) = \frac{\partial g}{\partial x}(x(T)). More generally, if the terminal state x(T) is constrained to a manifold defined by \psi(x(T)) = 0 with \psi: \mathbb{R}^n \to \mathbb{R}^{n-m}, the costate at T must be orthogonal to the manifold's tangent space, expressed as \lambda(T) = \nu^\top \frac{\partial \psi}{\partial x}(x(T)) for some Lagrange multiplier vector \nu \in \mathbb{R}^{n-m}. These conditions arise from the need to satisfy the principle of optimality at the boundary, analogous to variational transversality but adapted to controlled systems via the Hamiltonian framework. Several key cases illustrate the transversality condition's role. For a free endpoint with no terminal cost (g \equiv 0), the condition simplifies to \lambda(T) = 0, implying the costate vanishes at the horizon's end. When the terminal time T is fixed but x(T) is free with a nonzero g, the full form \lambda(T) = g_x(x(T)) applies directly. If the terminal time T is variable (free), an additional scalar condition emerges: H(T) + \frac{\partial g}{\partial t}(x(T), T) = 0, which balances the Hamiltonian's value against any explicit time dependence in the terminal cost; if g is time-independent, this reduces to H(T) = 0. These cases ensure the overall boundary conditions are consistent with the maximum principle's state-costate symmetry. A representative example is the finite-horizon linear quadratic regulator (LQR) with free final state and no terminal cost, minimizing \int_{t_0}^T (x^\top Q x + u^\top R u)\, dt subject to \dot{x} = A x + B u, where Q \geq 0, R > 0 are weighting matrices. The Hamiltonian is H = \lambda^\top (A x + B u) - (x^\top Q x + u^\top R u), leading to costate dynamics \dot{\lambda} = -A^\top \lambda + 2 Q x and optimal control u^* = -R^{-1} B^\top \lambda. Assuming a quadratic costate form \lambda(t) = 2 P(t) x(t), substitution yields the Riccati equation \dot{P} = -A^\top P - P A + P B R^{-1} B^\top P - Q with terminal boundary P(T) = 0 from the transversality condition \lambda(T) = 0. This results in time-varying feedback u^*(t) = -R^{-1} B^\top P(t) x(t), where the vanishing terminal costate reflects the absence of final-state penalties.^[16]

Infinite-Horizon Transversality

In infinite-horizon optimal control problems, the agent seeks to minimize the discounted infinite integral of the running cost functional

\int_0^\infty e^{-\rho t} L(x(t), u(t)) \, dt

subject to the state equation

\dot{x}(t) = f(x(t), u(t)), \quad x(0) = x_0,

where \rho > 0 denotes the positive discount rate, x(t) \in \mathbb{R}^n is the state vector, u(t) \in U \subseteq \mathbb{R}^m is the control vector, and L and f are assumed sufficiently smooth to ensure existence of optimal solutions.^[17] The Pontryagin maximum principle provides necessary conditions for optimality, including maximization of the discounted Hamiltonian H(t, x, u, \lambda) = e^{-\rho t} [L(x, u) + \lambda \cdot f(x, u)] with respect to u and the costate equation \dot{\lambda}(t) = -\frac{\partial H}{\partial x}(t, x(t), u(t), \lambda(t)). To close the system and guarantee convergence of the objective to a finite value, a transversality condition must hold at the infinite horizon: under appropriate bounded growth assumptions on the trajectories (e.g., |x(t)| \leq K e^{\gamma t} for some K > 0, \gamma < \rho),

\lim_{t \to \infty} e^{-\rho t} \lambda(t) \cdot x(t) = 0,

or, more stringently in cases with stricter boundedness, \lim_{t \to \infty} \lambda(t) = 0. This condition is derived by integrating the discounted Hamiltonian along the optimal trajectory and requiring the boundary term at infinity to vanish, ensuring the integral remains well-defined and the solution does not explode.^[17]^[18] The infinite-horizon transversality condition bears a direct analogy to the no-Ponzi scheme restriction prevalent in economic dynamics, which prohibits trajectories that allow infinite arbitrage or perpetual debt rollover without repayment. By mandating that the discounted product of the costate (shadow price) and state vanishes in the limit, it enforces sustainability, preventing scenarios where the agent could exploit discounting to accumulate unbounded value without finite cost, thus aligning the mathematical optimum with economically feasible paths.^[19] A representative illustration is the cake-eating problem, a canonical resource depletion model where the objective is to maximize

\int_0^\infty e^{-\rho t} u(c(t)) \, dt

subject to the resource dynamics \dot{S}(t) = -c(t), S(0) = S_0 > 0, S(t) \geq 0, with u(\cdot) a strictly concave, increasing utility function and c(t) \geq 0 the extraction (consumption) rate. The associated current-value Hamiltonian is H(S, c, \psi) = u(c) + \psi (-c), yielding the conditions \frac{\partial H}{\partial c} = 0 (so u'(c(t)) = \psi(t)) and \dot{\psi}(t) = \rho \psi(t) - \frac{\partial H}{\partial S}(t) = \rho \psi(t). The transversality condition \lim_{t \to \infty} \psi(t) S(t) e^{-\rho t} = 0 (equivalent to \lim_{t \to \infty} \lambda(t) S(t) = 0 in present-value terms, where \lambda(t) = e^{-\rho t} \psi(t)) rules out over-extraction paths that deplete the stock too slowly or rapidly, ensuring the optimal solution features declining marginal utility matched to the discount rate, with full asymptotic depletion S(\infty) = 0.^[20]

Applications

Economic Growth Models

In economic growth models, the transversality condition plays a crucial role in ensuring the sustainability of optimal capital accumulation paths over infinite horizons. The seminal Ramsey-Cass-Koopmans model exemplifies this application, where the social planner maximizes the discounted integral of utility from consumption, \int_0^\infty u(c(t)) e^{-\rho t} dt, subject to the capital accumulation constraint \dot{k}(t) = f(k(t)) - \delta k(t) - c(t), with k denoting capital per capita, c consumption per capita, f(k) the production function, \delta the depreciation rate, and \rho the discount rate. The transversality condition in this framework requires that \lim_{t \to \infty} e^{-\rho t} \mu(t) k(t) = 0, where \mu(t) is the shadow price of capital, derived from the Hamiltonian. This condition prevents economically infeasible outcomes, such as infinite debt accumulation, by ruling out paths where capital grows explosively without bound. The transversality condition, combined with the Euler equation governing the evolution of the shadow price \dot{\mu} = \rho \mu - f'(k), uniquely selects the stable trajectory that converges to the steady-state capital stock. Without it, multiple solutions to the differential equations could exist, including unstable ones that diverge; the condition eliminates these, ensuring the optimal path aligns with long-run economic equilibrium where marginal product of capital equals the discount rate, f'(k^*) = \rho + \delta. This convergence property underscores the condition's necessity for well-behaved dynamics in neoclassical growth theory. Historically, David Cass (1965) and Tjalling C. Koopmans (1965) formalized the use of the transversality condition to prove the optimality of decentralized competitive equilibria in the neoclassical growth model, demonstrating that they achieve the social planner's solution under perfect foresight. Their work established the condition's role in linking the optimal savings rate to the "golden rule" capital stock, which maximizes steady-state consumption per capita, f'(k_g) = \delta. This insight resolved earlier ambiguities in Ramsey's 1928 framework and solidified the model's foundations for analyzing long-term growth and policy. In endogenous growth models like the AK model, where output is linear in capital (f(k) = Ak), the transversality condition simplifies to imply balanced growth only if the discount rate equals the marginal product, \rho = A - \delta. This ensures constant capital per capita along the optimal path, avoiding explosive growth that would violate intertemporal budget constraints, and highlights the condition's adaptability to models without diminishing returns.

Natural Resource Extraction

In non-renewable resource economics, the transversality condition plays a central role in determining optimal depletion paths over infinite horizons, as formalized in the Hotelling rule framework using optimal control theory. The problem is to maximize the present value of profits from extraction: \max \int_0^\infty e^{-\rho t} \pi(p(t), q(t)) \, dt, subject to the resource stock dynamics \dot{S}(t) = -q(t), with initial stock S(0) = S_0 > 0 and S(\infty) = 0, where \rho > 0 is the discount rate, q(t) is the extraction rate, p(t) is the price, and \pi denotes profits net of extraction costs.^[21] The current-value Hamiltonian is H = \pi(p(t), q(t)) + \lambda(t) (-q(t)), where \lambda(t) is the shadow price of the resource stock. Maximization yields the first-order condition \frac{\partial H}{\partial q} = 0, implying \frac{\partial \pi}{\partial q} = \lambda(t). The co-state equation is \dot{\lambda} = \rho \lambda - \frac{\partial H}{\partial S} = \rho \lambda, since the Hamiltonian does not explicitly depend on S, leading to the Hotelling rule: \dot{\lambda} = \rho \lambda, or the shadow price rises at the discount rate.^[21]^[22] The transversality condition for this infinite-horizon problem is \lim_{t \to \infty} e^{-\rho t} \lambda(t) S(t) = 0, ensuring that the discounted value of the remaining resource stock approaches zero as time goes to infinity.^[21] This condition prevents backloading extraction—delaying depletion indefinitely to exploit the rising shadow price—by requiring full exhaustion of the stock over the infinite horizon. It interprets the resource's in-situ value as approaching zero in present-value terms, avoiding scenarios where positive stock persists forever with positive marginal value. Combined with a no-Ponzi-game condition that bounds debt accumulation (preventing perpetual financing of extraction through borrowing), the transversality condition ensures sustainable depletion paths without infinite debt buildup from resource sales.^[23] A key insight from this setup is that, in a competitive equilibrium, the resource price net of marginal extraction costs rises at the discount rate, as the shadow price \lambda(t) equals this net price.^[22] The transversality condition rules out leaving the resource in the ground indefinitely, as that would violate the zero terminal value requirement unless \rho = 0, enforcing economic efficiency by aligning extraction with intertemporal opportunity costs. For example, in an oil extraction model without backstop technologies, the condition implies complete depletion of reserves over infinite time if \rho > 0, with extraction rates declining as the net price rises exponentially; this contrasts with finite-horizon models incorporating backstops (unlimited substitute fuels), where extraction may halt prematurely upon reaching the backstop price threshold.^[21]

Mathematical Properties

Necessity and Sufficiency

In the calculus of variations, the transversality condition arises as a necessary optimality criterion for problems with variable endpoints. To establish necessity, consider a functional J = \int_{x_0}^{x_1} F(x, y, y') \, dx where the endpoint (x_1, y(x_1)) lies on a curve \psi(x), and y satisfies the Euler-Lagrange equation interiorly. Perturb the extremal curve y(x) by a small variation h(x) such that the perturbed endpoint remains on \psi(x), ensuring the variation satisfies \delta y_1 = \psi'(x_1) \delta x_1. The first variation \delta J must vanish for optimality: after integration by parts, the boundary term at x_1 yields [F - y' F_{y'}] \delta x_1 + F_{y'} \delta y_1 = 0. Substituting the endpoint constraint gives the transversality condition F + (\psi' - y') F_{y'} = 0 at the endpoint, confirming it is required for stationarity.^[6] For infinite-horizon optimal control problems, such as maximizing \int_0^\infty e^{-\rho t} f(x(t), u(t)) \, dt subject to \dot{x} = g(x, u) and x(0) = x_0, necessity follows from perturbation arguments on admissible paths. Consider a perturbed trajectory (x(t), u(t)) close to the optimal (x^*(t), u^*(t)), and define the difference in discounted value D_T = \int_0^T e^{-\rho t} [f(x, u) - f(x^*, u^*)] \, dt \leq 0 for optimality up to finite T. Using the Hamiltonian H = f + \lambda g and its adjoint \dot{\lambda} = -\frac{\partial H}{\partial x}, integration by parts on the discounted integral transforms the boundary terms: \int_0^T e^{-\rho t} (\lambda \dot{x} + \dot{\lambda} x) \, dt = [\lambda x e^{-\rho t}]_0^T + \rho \int_0^T e^{-\rho t} \lambda x \, dt. As T \to \infty, for D_T \leq 0 to hold and the integral to converge, the transversality condition \lim_{T \to \infty} e^{-\rho T} \lambda(T) \cdot x^*(T) = 0 must obtain, ensuring the variation \delta J \geq 0.^[24] Sufficiency of the transversality condition, when combined with the Euler-Lagrange equations or Pontryagin's maximum principle, holds in convex problems. For instance, in linear-quadratic systems or when the Hamiltonian is jointly concave in (x, u), satisfaction of the necessary conditions—including transversality—implies a global optimum, as the second variation is non-negative and boundary terms vanish appropriately. An extension of the Weierstrass theorem provides sufficiency for strong minima in variational problems if the Weierstrass excess function E(x, y, y'; \bar{y}') = F(x, y, \bar{y}') - F(x, y, y') - F_{y'}(x, y, y') (\bar{y}' - y') \leq 0 along the extremal, alongside transversality, ensuring no better nearby path exists. In reduced-form economic models, Kamihigashi's elementary proof shows that transversality, derived via perturbations \{x_t^*, \lambda x_{t+1}^*\} for \lambda \in [0,1), suffices for local optimality under concavity and differentiability assumptions.^[25] However, sufficiency fails in non-convex cases, where necessary conditions may identify local but not global optima, or spurious solutions; for example, in problems with non-concave Hamiltonians, transversality alone cannot rule out oscillatory paths that violate global optimality. Limitations arise without growth bounds or discounting: in undiscounted infinite-horizon models, perturbations can yield non-convergent integrals, invalidating the condition and allowing divergent optimal paths.^[26]

Generalizations and Extensions

In stochastic optimal control problems governed by Itô processes, the transversality condition is extended to incorporate uncertainty through expectation operators and diffusion effects in the underlying Hamilton-Jacobi-Bellman equation. For infinite-horizon discounted formulations, it typically requires \mathbb{E}\left[ \lim_{t \to \infty} e^{-\rho t} \lambda(t)^\top x(t) \right] = 0, where \lambda(t) denotes the adjoint process, x(t) the state vector, \rho > 0 the discount rate, and \mathbb{E}[\cdot] the expectation under the optimal control policy. This ensures the discounted marginal value of the state approaches zero asymptotically, preventing arbitrage opportunities in the presence of stochastic shocks. The condition arises from the stochastic Pontryagin maximum principle applied to forward-backward stochastic differential equations, with diffusion terms influencing the adjoint dynamics.^[27]^[28] For optimal control problems with state constraints, the transversality condition is generalized using Lagrange multipliers associated with the constraint manifolds, allowing the adjoint variable to exhibit jumps at points where the state enters or exits the constraint set. These multipliers, often non-negative and satisfying complementarity slackness, adjust the boundary conditions to maintain optimality on the feasible set, for example, at an interior junction time t_k, \lambda(t_k^+) = \lambda(t_k^-) + \mu \nabla g(x(t_k)) where g(x(t_k)) = 0 and \mu \geq 0.^[29] This framework handles pure or mixed state constraints, ensuring the Hamiltonian remains maximized along constrained arcs. In multipoint boundary value problems, including those for periodic orbits, transversality conditions are applied at multiple endpoints to enforce closure and optimality across the trajectory segments. For periodic solutions in autonomous systems, the conditions include periodicity of both state and adjoint, x(0) = x(T) and \lambda(0) = \lambda(T), combined with Hamiltonian periodicity H(0) = H(T), which together characterize locally optimal closed orbits without fixed terminal times. These ensure the trajectory returns to the initial manifold transversally while satisfying the necessary conditions of the Pontryagin maximum principle.^[30] Distinctions between discounted and undiscounted cases lead to varied transversality formulations, particularly for autonomous systems lacking a discount factor \rho. In undiscounted infinite-horizon problems, a common condition is \limsup_{T \to \infty} \lambda(T)^\top (x^*(T) - x(T)) = 0 for admissible perturbations x near the candidate optimal x^*, which helps ensure overtaking optimality where finite-horizon approximations may not converge uniformly. This weaker requirement replaces the stricter zero-limit of discounted settings, avoiding infeasibility in long-run steady states.^[24] Modern extensions include applications in differential games for non-cooperative multi-agent settings, as pioneered by Isaacs in 1965, where transversality conditions on the terminal surface or capture set determine the value function via coupled adjoint equations for each player, ensuring saddle-point equilibria. Additionally, numerical methods like shooting algorithms enforce transversality by iteratively adjusting initial costates to match boundary residuals, with multiple shooting enhancing convergence for high-dimensional or singular problems through parallel arc integration.^[31]^[32]^[33]

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[PDF] The Calculus of Variations
Dec 1, 2022 · This is the transversality condition of §9, at P2(t0). (iii) If fy0y0 (x, y, y0) 6= 0 along Et0 , then a further necessary condition follows ...
[7]
Calculus of variations : Bliss, Gilbert Ames, 1876-1951
Mar 14, 2019 · Calculus of variations. by: Bliss, Gilbert Ames, 1876-1951. Publication date: 1925. Topics: Calculus of variations. Publisher: Chicago, Ill ...Missing: transversality | Show results with:transversality
[8]
[PDF] The Maximum Principle of Pontryagin in control and in optimal control
May 23, 2006 · The final ingredient in the Maximum Principle is the transversality conditions. First of all, in the differential geometric formulation one ...
[9]
[PDF] Paul Samuelson's ways to macroeconomic dynamics - EconStor
May 18, 2019 · Samuelson's (1947) approach to macroeconomic dynamics as separate from optimization remained essentially unchallenged until the mid 1970s, ...Missing: transversality | Show results with:transversality
[10]
[PDF] Optimal Control Theory with Applications in Economics
For example, Arrow (1968) and Arrow and Kurz (1970a) provided an early ... dynamic optimization problem for a system with unconstrained states. Given.
[11]
[PDF] cvoc.pdf - Daniel Liberzon
Aug 9, 2011 · ... textbooks on calculus of variations and optimal control systems ... The transversality condition (7.36) is completely analogous to the ...
[12]
[PDF] CALCULUS OF VARIATIONS and TENSOR CALCULUS - OSU Math
Sep 22, 2019 · ... (y) (x,y)=(x1,y1). = 0. Fy′ + (F − y′Fy′ )Φ′(y) (x,y)=(x0,y0). = 0. One must use this kind of transversality condition if there are many degrees ...
[13]
[PDF] LECTURE 6 6. The Linear Quadratic Regulator 6.1. Fixed-time, free ...
and by the transversality condition for the free-endpoint problem p∗(t1) = −2Mx∗(t1). Since H is continuously differentiable in u, the Hamiltonian ...
[14]
[PDF] Infinite Horizon Optimal Control: Transversality Conditions ... - HAL
Aug 18, 2022 · Let us underline that in the finite horizon setting, the transversality condition at final time is one of ingredients of the maximum principle.<|control11|><|separator|>
[15]
[PDF] Transversality Conditions in Infinite Horizon Models
L = ε st { V[c(t),m(t)] + a(t)[f[c(t),m(t)] - m(t+1)] +. Σ t=1 a(t)[u(t) - c(t)] + p(t) [c(t) - e(t)]} + 1(0)(m − m(0)).<|control11|><|separator|>
[16]
[PDF] NECESSARY AND SUFFICIENT CONDITIONS FOR DYNAMIC ...
We characterize the necessary and sufficient conditions for optimality in discrete-time, infinite-horizon optimization problems with a state space of finite ...Missing: Samuelson 1947
[17]
[PDF] The Calculus of Variations and Optimal Control in Economics and ...
The aim of the series is to cover topics in economics, mathematical economics, and econometrics, at a level suitable for graduate students or final year.
[18]
[PDF] Optimal control theory with applications to resource and ... - EconStor
Jan 20, 2016 · From (26) it is clear that if extraction stops while 6= 0, the transversality condition (27) will be violated. Similarly, if becomes negative, ...
[19]
[PDF] The Economics of Exhaustible Resources - Regulations.gov
The Economics of Exhaustible Resources. Author(s): Harold Hotelling. Source: Journal of Political Economy , Apr., 1931, Vol. 39, No. 2 (Apr., 1931), pp. 137 ...
[20]
[PDF] Sustainability and substitution of exhaustible natural resources. How ...
... resources. (and earning capital gains because of increases in the resource price). The transversality condition and the No-Ponzi-game condition require:.
[21]
[PDF] Lecture Notes 8: Dynamic Optimization Part 2: Optimal Control
The Malinvaud transversality condition is that p(T)x∗(T) → 0 as T → ∞. When this Malinvaud transversality condition is satisfied, evidently lim sup. T→∞ p(T) ...
[22]
A simple proof of the necessity of the transversality condition
This note provides a simple proof of the necessity of the transversality condition for the differentiable reduced-form model. The proof uses only an elemen.
[23]
Value Functions and Transversality Conditions for Infinite-Horizon ...
Synthesizing these results, necessary and sufficient conditions for optimality are provided for the convex case. In particular, the role of the transversality ...
[24]
Necessity of the transversality condition for stochastic models with ...
This paper shows that the standard transversality condition (STVC) is necessary for optimality in stochastic models with bounded or ...Missing: seminal | Show results with:seminal
[25]
[PDF] Necessity of Transversality Conditions for Stochastic Problems
This paper establishes (i) an extension of Michel's (1990, Theorem 1) necessity result to an abstract reduced-form model, (ii) a generalization of the ...Missing: seminal | Show results with:seminal
[26]
The Pontryagin Maximum Principle and Transversality Conditions ...
This paper gives a survey of the various forms of Pontryagin's maximum principle for optimal control problems with state variable inequality constraints. The ...
[27]
[PDF] Interplanetary transfers with low consumption using the properties of ...
Feb 16, 2017 · periodic orbits to satisfy the necessary transversality conditions given by the Pontryagin ... optimal control problem of orbital transfer ...
[28]
Differential Games - Rufus Isaacs - Google Books
Definitive work draws on game theory, calculus of variations, and control theory to solve an array of problems: military, pursuit and evasion, athletic ...Missing: transversality | Show results with:transversality
[29]
[PDF] Differential Games : Lecture notes on the Isaacs-Breakwell Theory
˙ λ2. = −λ1. The transversality condition (11) yields here that on the line x1 = γ, λ2 = 0. The value has to be positive in the playing region x1 < γ, hence.
[30]
[PDF] Survey of Numerical Methods for Optimal Control - Anil Rao
A survey of numerical methods for optimal control is given. The objective of the article is to describe the major methods that have been developed over the ...