The Mathematical Theory of Optimal Processes

By M. L. Chambers. 1965

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Citations in the corpus

FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models (2018, background): “For any scalar loss function which operates on the solution to an initial value problem then Pontryagin (1962) shows that its derivative takes the form of another initial value problem The quantity − ∂L / ∂ z ( t ) is known as the adjoint state of the ODE.”
Neural Ordinary Differential Equations (2018, positive): “We treat the ODE solver as a black box, and compute gradients using the adjoint method (Pontryagin et al., 1962).”
Deep learning in neural networks: An overview (2014, methodology): “For example, certain methods of syntactic pattern recognition (Fu, 1977) such as grammar induction discover hierarchies of formal rules to model observations.”
Normalizing Flows for Probabilistic Modeling and Inference (2019, background): “a result known widely as the adjoint sensitivity method (Pontryagin, 1962).”
A Survey of Motion Planning and Control Techniques for Self-Driving Urban Vehicles (2016, background): “Vehicle motion must then be planned and regulated to accomplish driving tasks and while respecting the constraints introduced by the selected model.”
Data-Efficient Reinforcement Learning with Probabilistic Model Predictive Control (2017, background): “With these results and the deﬁnition of the control-Hamiltonian 13 we can now state the PMP for the control system ( Z , f MM , U ) as follows: Theorem 1.”
Motion Planning (2008, unknown): “We recall in Section 2 its basic version which we will use, and refer to classical books in control theory, like [1], [4], and [7], for its quite delicate proof.”
Solving optimal control problems of Fredholm constraint optimality via the reproducing kernel Hilbert space method with error estimates and convergence analysis (2019, unknown): “regarding to the accompanying dynamical system d dt x t ð Þ þ Dt x t ð Þ 1⁄4 u t ð Þ þ t(2); x 0 ð Þ 1⁄4 0: (8)<: (50)”
Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method (2006, methodology): “II is to apply the calculus of variations and Pontryagin’s maximum principle [24] to obtain first-order necessary conditions for optimality [2].”
Low-Thrust Transfers in the Earth-Moon System, Including Applications to Libration Point Orbits (2010, background): “(40) is always satisfied by defining P = Pmax.”
A uniform Tauberian theorem in optimal control (2010, unknown): “Finite horizon problem of optimal control have been studied intensively since the pioneer work of Stekhov, Pontryagin, Boltyanskii [28], Hestenes [18], Bellman [9] and Isaacs [19,20] during the cold war - see for instance [7,23,24] for major references, or [14] for a short, clear introduction.”
Optimal Observation of the One-dimensional Wave Equation (2013, methodology): “According to the Pontryagin Maximum Principle (see [22]), if a is optimal then there exist real numbers(2) (py, p1, .”
Needle Variations in Infinite-Horizon Optimal Control (2014, background): “To our knowledge for the first time a version of the maximum principle for the infinitehorizon optimal control problem was proved in [16] in the case when the improper integral utility functional converges and the optimal trajectories satisfy additional boundary constraint limt→∞ x∗(t) = x1, where x”
On L1‐minimization in optimal control and applications to robotics (2006, background): “The classical Pontryagin’s minimum principle [1] deals with optimal control problems where all occurring functions are continuous with respect to the control and continuously differentiable with respect to the state.”
Spectral Algorithm for Pseudospectral Methods in Optimal Control (2008, background): “Alternatively, this issuemay cast suspicion on theoretical results on convergence if it can be proved that the problem does indeed have a solution.”
Differential inclusions and the control of forest fires (2007, background): “22) will then follow from a direct application of the Pontryagin maximum principle [2].”