A BSTRACT . We consider the isometry group of the infinitedimensional separable hyperbolic space with its Polish topology. This topology is given by the pointwise convergence. For non- locally compact Polish groups, some striking phenomena like automatic continuity or extreme amenability may happen. Our leading idea is to compare this topological group with usual Lie groups on one side and with non-Archimedean infinitedimensional groups like S ∞ , the group of all permutations of a countable set on the other side. Our main results are
algebra of the group if the group is finite, or in the universal enveloping algebra of the Lie algebra if one deals with a Liegroup. The central elements form a com- mutative algebra that is being mapped to the algebra of functions on the labels, i.e., on partitions or their relatives. The value of the function corresponding to a central element at a representation label is the (scalar) value of this element in that representation.
| f ( g 0 , . . . , g n )| .
When G is a locally compact group, Burger and Monod [ BM99 ] defined the continu- ous bounded cohomology H n cb ( G, R ) of G and showed that, in degree 2, the comparison map c : H 2 cb ( G, R ) → H 2 c ( G, R ) is an isomorphism. Here H 2 c ( G, R ) denotes the continuous co- homology of G (a standard text about continuous cohomology is [ BW00 ]). The result of Burger-Monod allows to give a complete description of H 2 cb ( G, R ) in case of semisimple Lie groups with finite center: the continuous cohomology H 2 c ( G, R ) can be identified with the vector space of G-invariant differential fom Ω 2 (X , R ) G where X is the symmetric space as- sociated to G. In particular, for a simple Liegroup G of non-compact type and finite center, the second continuous cohomology H 2 cb ( G, R ) is equal to Rκ cb G if X is a Hermitian symmet- ric space (and κ cb G is then the bounded Kähler class, see below), and vanishes otherwise. In general H 2 cb ( G, R ) is generated by the bounded Kähler classes of the Hermitian factors of X . Of course, when G is a discrete group, the continuity assumption is void, and so the contin- uous (bounded) cohomology H 2 c ( b ) ( G, R ) agrees with the absolute (bounded) cohomology H 2 ( b ) ( G, R ) , in general not much is known about the natural maps H 2 c ( b ) ( G, R ) → H 2 ( b ) ( G, R ) induced by the inclusion of continuous (bounded) cochains in absolute (bounded) cochains, nor about the absolute bounded cohomology of Lie groups regarded as discrete groups. 5.2. The bounded Kähler class. We now turn our attention to the bounded cohomology of the groups G of isometries of the infinitedimensional Hermitian symmetric spaces X intro- duced in Section 2 . Since such groups G are not locally compact, there is no well-established theory of continuous bounded cohomology, therefore we will just work with the bounded cohomology H 2 b ( G, R ) . If X has finite rank, we can use the Kähler form to define a class in the bounded cohomology of G, precisely as in the finite dimensional case:
In an infinitedimensional setting, that is for economies with infinitely many commodities, the result first appeared in a seminal paper of Debreu (1954)  where the commodity space is a topological vector space. Besides convexity hypothesis, a key assumption of Debreu’s result is that the total production set of the economy has a nonempty interior, an assumption strongly related by him with free-disposal in commodities, which clearly reduces the application of the result to ordered topological vector commodity spaces whose positive orthant has a nonempty interior. A simple look to Debreu’s proof, based on the Hahn-Banach theorem, allows for noticing that the interiority assumption could be likewise done for anyone of the preferred sets. However, given the assumptions currently used for guaranteeing the existence of Pareto optimal allocations, such an assumption would equally be suited for commodity spaces whose positive cone has a nonempty interior but, as previously, would prevent any application of the result to a number of commodity spaces which have been found of economic interest. This limitation explains why, thirty two years after, Debreu’s result was revisited by Mas-Colell (1986)  who replaces Debreu’s interiority condition by the so-called properness assumptions on preferences and production, at the cost of strengthening the assumptions made on the commodity space which is assumed in  to be a topological vector lattice and on the consumption sets restricted to coincide with the positive orthant. As in Debreu , the weak optimum is supported by a non-zero, continuous linear functional p called “valuation functional”. Moreover, p(ω) > 0 if the total initial endowment, ω > 0, is the common properness vector.
Usually, when one develops the theory of the horofunction boundary, one makes the assumption that the space is proper, that is, that closed balls are compact. For normed spaces and Hilbert geometries, this is equivalent to the dimension being finite. To deal with infinitedimensional spaces, we are forced to extend the framework. For example, we must use nets rather than sequences. In section 2, we reprove some basic results concerning the horofunction boundary in this setting. We study the boundary of normed spaces in section 3. In this and later sections, We make extensive use of the theory of affine functions on a compact set, including some Choquet Theory. To demonstrate the usefulness of the horofunction boundary, we give a short proof of the Masur–Ulam theorem in section 4. As an example, we determine explicitly the Busemann points in the boundary of the important Banach space (C(K), || · || ∞ ) in section 5. Important to
PRINCIPLES FOR DISCRETE-TIME PROBLEMS
MOHAMMED BACHIR AND JO¨ EL BLOT
Abstract. The aim of this paper is to provide improvments to Pontryagin principles in infinite-horizon discrete-time framework when the space of states and of space of controls are infinite-dimensional. We use the method of re- duction to finite horizon and several functional-analytic lemmas to realize our aim.
This concept is widely used in the literature.
Optimal control: dynamic programming and verification theorems .
As in the study of finite-dimensional stochastic (and non-stochastic) optimal control problem, the dynamic programming approach connects the study of the minimization problem with the analysis of the related Hamilton-Jacobi-Bellman (HJB) equation: given a solution of HJB and a certain number of hypotheses, the optimal control can be found in feedback form (i.e. as a function of the state) through a so called verification theorem. The idea is to identify a solution of the HJB equation with the value function of the control problem. When the state equation of the optimal control problem is an infinitedimensional stochastic evolution equation, the related HJB is of second order and infinitedimensional. The simplest procedure for establishing a verification theorem consists in considering the regular case where the solution is assumed to have all the regularity needed to give meaning to all the terms appearing in the HJB in the classical sense: it needs to be C 1 in the time variable and C 2 in
its differential is complex linear follows from the holomorphy of f ∧ .
In the following, we say that a Liegroup structure on O(M, K) is compatible with evalua- tions if it satisfies the assumptions of the preceding proposition. As in the real case, we obtain: Corollary III.7. Under the assumptions of the preceding theorem, there exists at most one regular complex Liegroup structure on the group O(M, K) which is compatible with evaluations. Theorem III.8. Let M be a connected complex manifold and K a complex regular Liegroup. Assume that the subset δ(O ∗ (M, K)) is a complex submanifold of Ω 1 h (M, k) and endow O ∗ (M, K)
Absolute humanoid localization and mapping based on IMU Liegroup and fiducial markers
Mederic Fourmy † , Dinesh Atchuthan † , Nicolas Mansard † , Joan Sol`a †∗ and Thomas Flayols †
Abstract— Current locomotion algorithms in structured (in- door) 3D environments require an accurate localization. The several and diverse sensors typically embedded on legged robots (IMU, coders, vision and/or LIDARS) should make it possible if properly fused. Yet this is a difficult task due to the hetero- geneity of these sensors and the real-time requirement of the control. While previous works were using staggered approaches (odometry at high frequency, sparsely corrected from vision and LIDAR localization), the recent progress in optimal estimation, in particular in visual-inertial localization, is paving the way to a holistic fusion. This paper is a contribution in this direction. We propose to quantify how a visual-inertial navigation system can accurately localize a humanoid robot in a 3D indoor environment tagged with fiducial markers. We introduce a the- oretical contribution strengthening the formulation of Forster’s IMU pre-integration, a practical contribution to avoid possible ambiguity raised by pose estimation of fiducial markers, and an experimental contribution on a humanoid dataset with ground truth. Our system is able to localize the robot with less than 2 cm errors once the environment is properly mapped. This would naturally extend to additional measurements corresponding to leg odometry (kinematic factors) thanks to the genericity of the proposed pre-integration algebra.
Keywords Stochastic calculus in Hilbert (Banach) spaces · Itô Formula Mathematics Subject Classification (2000) 60H05 · 60H15 · 60H30
Stochastic calculus and in particular Itô formula has been extended since long time ago from the finite to the infinite-dimensional case. The final result in infinite dimensions is rather similar to the finite- dimensional one except for some important details. One of them is the fact that unbounded (often linear) operators usually appears in infinite dimensions and Itô formula has to cope with them. If the Itô process X (t), taking values in a Hilbert space H, satisfies an identity of the form
In [Bigot et al., 2010], using N independent copies of the process X, we have proposed to construct an estimator of the covariance matrix Σ by expanding the process X into a dictionary of basis functions. The method in [Bigot et al., 2010] is based on model selection techniques by empirical contrast minimization in a suitable matrix regression model. This new approach to covariance estimation is well adapted to the case of low-dimensional covariance estimation when the number of replicates N of the process is larger than the number of observations points n. However, many application areas are currently dealing with the problem of estimating a covariance matrix when the number of observations at hand is small when compared to the number of parameters to estimate. Examples include biomedical imaging, proteomic/genomic data, signal processing in neurosciences and many others. This issue corresponds to the problem of covariance estimation for high-dimensional data. This problem is challenging since, in a high- dimensional setting (when n >> N or n ∼ N), it is well known that the sample covariance matrices
L 1 (R); it is a subset of the space C 0 (R) of continuous functions vanishing at infinity.
II.2 Definition of the notion of solution
Let us consider a new model, with the purpose to make a link between the results obtained in this paper and recent regularization by noise results for McKean-Vlasov equations obtained among others by [Jou97, MV, Lac18, CdRF, RZ]. There are some important changes with respect to the model (13) previously studied in this work. The main modification consists in adding a Brownian motion β, independent of w, in order to take benefit from some additional regularizing effect. In short, the role of β in the model below is to smooth out the (finite dimensional) space variable in the drift coefficient. Obviously, this comes in contrast with the role of the Brownian sheet w, the action of which is to mollify the velocity field in the measure argument, as made clear by Theorem 17. Of course, we know from the standard diffusive case (i.e. w ≡ 0 and
Int K ̸= ∅, we work in the Banach space of the continuous functions. At ﬁrst the zero- obstacle problem is investigated, namely, we require the positivity of the solutions. Example 3.3 (A one-dimensional heat equation). To model the heat ﬂux in a cylindrical bar, with perfectly insulated lateral surface and whose length is much larger than its cross- section, a one-dimensional equation is introduced. For u = u(t, s), t, s ∈ [0, 1] × [0, 1], let
The aim of this paper is to extend the theory devel- oped in the previous references by using IQCs as exposed in Scherer and Veenman (2018). In this context, a fil- ter providing the projection coefficients of the infinite- dimensional state is derived. Thanks to this filter, a new class of IQCs can be generated, not related to a specific class of PDEs. In that sense, the proposed result is adapt- able to many linear PDEs and it is promising to extend these ideas to other classes.
We first derive a general integral-turnpike property around a set for infinite-dimensional non-autonomous optimal control problems with any possible terminal state constraints, un- der some appropriate assumptions. Roughly speaking, the integral-turnpike property means that the time average of the distance from any optimal trajectory to the turnpike set con- verges to zero, as the time horizon tends to infinity. Then, we establish the measure-turnpike property for strictly dissipative optimal control systems, with state and control constraints. The measure-turnpike property, which is slightly stronger than the integral-turnpike property, means that any optimal (state and control) solution remains essentially, along the time frame, close to an optimal solution of an associated static optimal control problem, except along a subset of times that is of small relative Lebesgue measure as the time horizon is large. Next, we prove that strict strong duality, which is a classical notion in optimization, implies strict dissipativity, and measure-turnpike. Finally, we conclude the paper with several comments and open problems.
vides a 5×5 matrix representation for the orientation, velocity and position of an object in the 3-D space, a triplet we call “extended pose”. In this paper we build on this group to develop a theory to associate uncertainty with extended poses represented by 5×5 matrices. Our approach is particularly suited to describe how uncertainty propagates when the extended pose represents the state of an Inertial Measurement Unit (IMU). In particular it allows revisiting the theory of IMU preintegration on manifold and reaching a further theoretic level in this field. Exact preintegration formulas that account for rotating Earth, that is, centrifugal force and Coriolis force, are derived as a byproduct, and the factors are shown to be more accurate. The approach is validated through extensive simulations and applied to sensor- fusion where a loosely-coupled fixed-lag smoother fuses IMU and LiDAR on one hour long experiments using our experimental car. It shows how handling rotating Earth may be beneficial for long-term navigation within incremental smoothing algorithms.