Endpoint Constraints Research Articles

Trajectory shaping guidance for impact angle control of planetary hopping robots

This paper presents a novel optimal trajectory-shaping control concept for a planetary hopping robot. The hopping robot suffers from uncontrolled in-flight and undesired after-landing motions, leading to a position drift at landing. The proposed concept thrives on the Generalized Vector Explicit (GENEX) guidance, which can generate and shape the optimal trajectory and satisfy the end-point constraints like the impact angle of the velocity vector. The proposed concept is used for a thruster-based hopping robot, which achieves a range of impact angles, reduces the position drift at landing due to the undesired in-flight and after-landing motions, and handles the error in initial hopping angles. The proposed approach’s conceptual realization is illustrated by lateral acceleration generated using thruster orientation control. Extensive simulations are carried out on horizontal and sloped surfaces with different initial and impact angle conditions to demonstrate the effect of impact angle on the position drift error and the viability of the proposed approach.

Avoidance of the Lavrentiev gap for one-dimensional non-autonomous functionals with constraints

Abstract Consider a positive functional F ⁢ ( y ) := ∫ t T L ⁢ ( s , y ⁢ ( s ) , y ′ ⁢ ( s ) ) ⁢ 𝑑 s {F(y):=\int_{t}^{T}L(s,y(s),y^{\prime}(s))\,ds} , defined on the space of Sobolev functions W 1 , p ⁢ ( [ t , T ] ; ℝ n ) {W^{1,p}([t,T];{\mathbb{R}}^{n})} , where p ≥ 1 {p\geq 1} . This functional is minimized among functions y that may satisfy one or both endpoint conditions. The Lagrangian L is allowed to assume the value + ∞ {+\infty} . In numerous applications minimizers may not be explicit or even may not exist. In such circumstances, it is crucial to know that the infimum of F can be approximated using a sequence of Lipschitz functions that meet the given boundary conditions. However, there are instances where this approximation is not feasible, even with polynomial Lagrangians that meet Tonelli’s existence conditions: this situation is referred to as the Lavrentiev phenomenon. Some results are present in the literature if one requires the Lipschitz approximations to preserve just one endpoint constraint or when the Lagrangian is finite valued. The present paper deals with problems with two endpoint constraints and Lagrangians that are allowed to take extended values. As a byproduct, our Lipschitz approximations may preserve given state constraints. The extended-valued case is challenging since the phenomenon may occur even when the Lagrangian is constant on its effective domain, whose topology becomes relevant. Once assumed that the Lagrangian is radial convex on the rays of the last variable, our findings offer new insights, even when the Lagrangian L is real-valued, autonomous, and there are no state constraints.

The maximum principle for lumped-distributed control systems.

This paper concerns the optimal control of lumped-distributed systems, an examplar of which is a mechanical system with rotating components connected by flexible rods. The underlying mathematical model is a controlled semilinear evolution equation, in which nonlinear terms involve a projection of the full state onto a finite dimensional subspace. We derive necessary conditions of optimality in the form of a maximum principle, for a problem formulation which involves pathwise and end-point constraints on the lumped components of the state variable. Perturbational methods are employed in the derivation, which allow for non-smooth data. A key step in the analysis is the reduction of the optimization problem with an infinite dimensional state space, to one with finite dimensional aspects. The computational implications of this reduction will be explored in future work. Necessary conditions for problems with state constraints and end-point constraints can only be achieved under rather strict compatibility hypotheses on the way the dynamics interact with the state constraint and end-point constraint sets, due the infinite dimensionality of the state space. This paper identifies a class of infinite dimensional problems, of engineering significance, for which necessary conditions of optimality can be derived, in the absence of any additional compatibility hypotheses.

Four-loop splitting functions in QCD – The quark-to-gluon case

We present the even-N moments N≤20 of the fourth-order (N3LO) contribution Pgq(3)(x) to the quark-to-gluon splitting function in perturbative QCD. These moments, obtained by analytically computing off-shell operator matrix elements for a general gauge group, agree with all known results, in particular with the moments N≤10 derived before from structure functions in deep-inelastic scattering. Using the new moments and the available endpoint constraints, we construct approximations for Pgq(3)(x) which improve upon those obtained from the lowest five even moments. The remaining uncertainties of this function are now practically irrelevant at momentum fractions x>0.1. The resulting errors of the convolution of Pgq at N3LO with a typical quark distribution are small at x≳10−3 and exceed 1% only at x≲10−4 for a strong coupling αs=0.2. The present results for Pgq(3)(x) should thus be sufficient for most collider-physics applications.

Necessary and sufficient conditions for one-dimensional variational problems with applications to elasticity

This paper deals with necessary and sufficient conditions for weak and strong minimizers of functionals Φ ( u ) = ∫ a b f ( x , u ( x ) , u ′ ( x ) ) d x , where u ∈ C 1 ( [ a , b ] , R N ) . We first derive conditions which are simpler than the known ones, and then apply them to several particular problems, including stability problems in the elasticity theory. In particular, we solve some open problems in [A. Majumdar, A. Raisch: Stability of twisted rods, helices and buckling solutions in three dimensions, Nonlinearity 27(2014), 2841–2867] by finding optimal conditions for the stability of a naturally straight Kirchhoff rod under various types of endpoint constraints.

CONDITIONAL COST FUNCTION AND NECESSARY OPTIMALITY CONDITIONS FOR INFINITE HORIZON OPTIMAL CONTROL PROBLEMS

Infinite horizon optimal control problem with general endpoint constraints is reduced to a family of standard problems on finite time intervals containing the value of the conditional cost of the phase vector as a terminal term. New version of the Pontryagin maximum principle containing an explicit characterization of the adjoint variable is obtained for the problem with a general asymptotic endpoint constraint. In the case of the problem with free final state this approach leads to a normal form version of the maximum principle formulated completely in the terms of the conditional cost function.

Four-loop splitting functions in QCD – The gluon-to-quark case

We have computed the even-N moments N≤20 of the gluon-to-quark splitting function Pqg at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. Our results, derived analytically for a general gauge group, agree with all results obtained for this function so far, in particular with the lowest five moments obtained via physical cross sections. Using our new moments and all available endpoint constraints, we construct approximations for the four-loop Pqg(x) that should be sufficient for a wide range of collider-physics applications. The N3LO corrections resulting from these and the corresponding quark-quark splitting functions lead to a marked improvement of the perturbative accuracy for the scale derivative of the singlet quark distribution, with effects of 1% or less at x≳10−4 at a standard reference scale with αs=0.2.

Necessary Conditions for the Optimality and Sustainability of Solutions in Infinite-Horizon Optimal Control Problems

The paper deals with an infinite-horizon optimal control problem with general asymptotic endpoint constraints. The fulfillment of constraints of this type can be viewed as the minimal necessary condition for the sustainability of solutions. A new version of the Pontryagin maximum principle with an explicitly specified adjoint variable is developed. The proof of the main results is based on the fact that the restriction of the optimal process to any finite time interval is a solution to the corresponding finite-horizon problem containing the conditional cost of the phase vector as a terminal term.

Time-optimal trajectory planning for autonomous parking of tractor-semitrailer vehicle based on piecewise Gauss pseudospectral method

For the autonomous parking problem of the tractor-semitrailer vehicle, a piecewise Gauss pseudospectral method is proposed as a trajectory planning scheme in this paper. Firstly, the trajectory planning problem is transformed into an optimal control problem by establishing a vehicle kinematics model for the low-velocity scene, the dynamic constraints including the physical constraints, the collision avoidance constraints, and the endpoint constraints. The optimal control problem are discretized into a nonlinear programming problem by piecewise Gauss pseudospectral method according to the movement area during parking process. The shortest time to complete parking is taken as the objective function. The Gauss pseudospectral method is further adopted to solve the nonlinear programming problem in the two parking stages. Three parallel parking conditions with different lengths of parking spaces and three vertical parking conditions with different widths of parking spaces are set for simulation verification based on Matlab/Simulink. The simulation results show that the proposed method can uniformly and effectively solve the trajectory planning problem of the tractor-semitrailer vehicle. Compared with the traditional pseudospectral method, a shorter parking time and a significantly higher convergence rate can be got with the method, which further indicates that the proposed method has good effectiveness and robustness.

Well-posedness of the shooting algorithm for control-affine problems with a scalar state constraint

We deal with a control-affine problem with scalar control subject to bounds, a scalar state constraint and endpoint constraints of equality type. For the numerical solution of this problem, we propose a shooting algorithm and provide a sufficient condition for its local convergence. We exhibit an example that illustrates the theory.

ON NORMALITY IN OPTIMAL CONTROL PROBLEMS WITH STATE CONSTRAINTS

A general optimal control problem with endpoint, mixed and state constraints is considered. The question of normality of the known necessary optimality conditions is investigated. Normality stands for the positiveness of the Lagrange multiplier λ0 corresponding to the cost functional. In order to prove the normality condition, an appropriate derivative operator for the state constraints is constructed, which acts in a specific Hilbert space and has the properties of surjection and continuity.

Automated control loop selection via multistage optimal control formulation and nonlinear programming

In this work, a novel approach based on the multistage optimal control formulation of the control loop selection problem is introduced. Currently, state-of-the-art approaches for controller loop design have been focused on data that yield only the pairings between input-output variables, and are not able to incorporate path and end-point constraints. Thus, they only produce the optimal loops for control purposes, without the simultaneous consideration of their optimal tuning. This formulation overcomes these drawbacks by producing an automated integrated solution for the task of control loop design, which also obviates the need for any form of combinatorial optimisastion to be used. To illustrate the procedure, as well as the advantages of the proposed scheme, different practical case studies are discussed and the results compared with those obtained with standard controller loop selection methods and their tuning. The results of the proposed approach show improved performance over previous methodologies found in the literature. Furthermore, the framework is extended to the selection of the control loops that must obey path and end-point constraints imposed by the underlying dynamical process. This task is usually difficult for classical methods, which violate them or exhibit underdamped response in some cases.

Four-loop splitting functions in QCD – The quark-quark case

We have computed the even-N moments N≤20 of the pure-singlet quark splitting function Pps at the fourth order of perturbative QCD via the anomalous dimensions of off-shell flavour-singlet operator matrix elements. Our results, derived analytically for a general gauge group, agree with all results obtained for this function so far, in particular with the lowest six even moments obtained via physical cross sections. Using these results and all available endpoint constraints, we construct approximations for Pps at four loops that should be sufficient for most collider-physics applications. Together with the known results for the non-singlet splitting function Pns+ at this order, this effectively completes the quark-quark contribution for the evolution of parton distribution at N3LO accuracy. Our new results thus provide a major step towards fully consistent N3LO calculations at the LHC and the reduction of the residual uncertainty in the parton evolution to the percent level.

Normality and second-order optimality conditions in state-constrained optimal control problems with bounded minimizers

A smooth optimal control problem with endpoint, mixed and state constraints is considered. The controllability matrix is constructed and the normality condition is presented as the condition of full rank for this matrix. The approach based on the controllability matrix allows one to examine the case of state-constrained control problems with fixed endpoints. As an application of normality, the second-order necessary optimality conditions are derived. The connection between different forms of second-order conditions is indicated.

Optimal control of higher-order retarded differential inclusions

The article investigates an optimal control problem described by retarded differential inclusions (DFIs) of higher order with endpoint constraints. In terms of the Euler-Lagrange type adjoint inclusions and the Hamiltonian, a sufficient optimality condition is derived for higher-order DFIs that have different forms in different time intervals depending on the delay parameter, which in the problem without delay, these two adjoint inclusions coincide. It is shown that the adjoint inclusion for the DFIs of the first order, defined in terms of locally conjugate mappings, coincides with the classical Euler-Lagrange inclusion. Thus, the obtained results are universal in the sense that sufficient optimality conditions can be formulated for a DFI of any order. At the end of the paper, problems with a high-order polyhedral DFIs and higher-order linear optimal control problems are considered, the optimality conditions of which are transformed into the Pontryagin maximum principle. Also, for high-order polyhedral optimization with delay, from the point of view of abstract economics, non-negative adjoint variables can be interpreted as the price of a resource.

SuperFusion: A Versatile Image Registration and Fusion Network with Semantic Awareness

Image fusion aims to integrate complementary information in source images to synthesize a fused image comprehensively characterizing the imaging scene. However, existing image fusion algorithms are only applicable to strictly aligned source images and cause severe artifacts in the fusion results when input images have slight shifts or deformations. In addition, the fusion results typically only have good visual effect, but neglect the semantic requirements of high-level vision tasks. This study incorporates image registration, image fusion, and semantic requirements of high-level vision tasks into a single framework and proposes a novel image registration and fusion method, named SuperFusion. Specifically, we design a registration network to estimate bidirectional deformation fields to rectify geometric distortions of input images under the supervision of both photometric and end-point constraints. The registration and fusion are combined in a symmetric scheme, in which while mutual promotion can be achieved by optimizing the naive fusion loss, it is further enhanced by the mono-modal consistent constraint on symmetric fusion outputs. In addition, the image fusion network is equipped with the global spatial attention mechanism to achieve adaptive feature integration. Moreover, the semantic constraint based on the pre-trained segmentation model and Lovasz-Softmax loss is deployed to guide the fusion network to focus more on the semantic requirements of high-level vision tasks. Extensive experiments on image registration, image fusion, and semantic segmentation tasks demonstrate the superiority of our SuperFusion compared to the state-of-the-art alternatives. The source code and pre-trained model are publicly available at https://github.com/Linfeng-Tang/SuperFusion.

Optimal control of differential inclusions with endpoint constraints and duality

The article considers a high-order optimal control problem and its dual problems described by high-order differential inclusions. In this regard, the established Euler–Lagrange type inclusion, containing the Euler–Poisson equation of the calculus of variations, is a sufficient optimality condition for a differential inclusion of a higher order. It is shown that the adjoint inclusion for the first-order differential inclusions, defined in terms of a locally adjoint mapping, coincides with the classical Euler–Lagrange inclusion. Then the duality theorems are proved.

An effective method in investigating structures of polytropic protoplanets formed via gravitational instability

In this article, we have reinvestigated the initial distribution of thermodynamic variables inside the protoplanets formed via gravitational instability having mass range 0.3−10MJ (1MJ=1.8986×1030 gm) by an embedded RKACeM(4,4) method assuming that the polytropic gas law holds in the protoplanets. The findings attained by our numerical experiments are recognized to be consistent with the results acquired through other notable investigations in this regard. Furthermore, the model is easily computable. The used method is found to be efficient in investing the structures of polytropic protoplanets in their initial stages in terms of accuracy, stability, computational cost, and solving endpoint constraints.

Necessary conditions for optimal control problems with sweeping systems and end point constraints

We generalize the Maximum Principle for free end point optimal control problems involving sweeping systems derived in [de Pinho MdR, Ferreira MMA, Smirnov GV. Optimal control involving sweeping processes. Set-Valued Var Anal. 2019;27(2):523–548] to cover the case where the constraints are time dependent and the end point is constrained to a set. As in [de Pinho MdR, Ferreira MMA, Smirnov GV. Optimal control involving sweeping processes. Set-Valued Var Anal. 2019;27(2):523–548], an ingenious smooth approximating family of standard differential equations plays a crucial role.

Problem of optimal control for bilinear systems with endpoint constraint

ABSTRACT In this work, we will investigate the question of optimal control for bilinear systems with constrained endpoint. The optimal control will be characterised through a set of unconstrained minimisation problems that approximate the former. Then a class of bilinear systems for which the optimal control can be expressed as a time-varying feedback law will be identified. Finally, applications to parabolic and hyperbolic partial differential equations are provided.