On uniform regularity and strong regularity

ABSTRACT We investigate uniform versions of (metric) regularity and strong (metric) regularity on compact subsets of Banach spaces, in particular, along continuous paths. These two properties turn out to play a key role in analyzing path-following schemes for tracking a solution trajectory of a parametric generalized equation or, more generally, of a differential generalized equation (DGE). The latter model allows us to describe in a unified way several problems in control and optimization such as differential variational inequalities and control systems with state constraints. We study two inexact path-following methods for DGEs having the order of the grid error O(h) and O(h2), respectively. We provide numerical experiments, comparing the schemes derived, for simple problems arising in physics. Finally, we study metric regularity of mappings associated with a particular case of the DGE arising in control theory. We establish the relationship between the pointwise version of this property and its counterpart in function spaces.


Introduction
We are going to investigate uniform (metric) regularity and strong (metric) regularity on compact subsets of Banach spaces of mappings which are defined as a sum of a single-valued (possibly non-smooth) mapping and a set-valued mapping with a (locally) closed graph. In the second section, we recall basic definitions from regularity theory and derive a result guaranteeing that a perturbed problem has a solution which is similar to the classical Lyusternik-Graves and Robinson theorem. Conditions ensuring uniform [strong] regularity along continuous paths are obtained as a corollary. Roughly speaking, by the word 'uniform' we mean that the constants as well as the size of neighbourhoods, appearing in the corresponding definitions, remain the same for a certain set of mappings and/or points. These properties turn out to be the key ingredients in the proofs of the non-smooth Robinson's inverse function theorem [1] and Lyusternik-Graves theorem for the sum a Lipschitz function and a set-valued mapping with closed graph [2]. To the best of our knowledge there is no self-contained study of these properties in the literature and the results are scattered here and there.
In the third section, we study two (inexact) path-following methods for a differential generalized equation (DGE), a model introduced in [3], which is a problem to find a pair of functions x : [0, T] → R n and u : , , u(t)) + F(u(t)), with a fixed T > 0, single-valued functions f : R n × R m → R d and g : R n × R m → R n , a set-valued mapping F : R m ⇒ R d , and a given initial state x I ∈ R n . This model allows us to describe in a unified way several problems in control and optimization such as differential variational inequalities and control systems with state constraints (see [3] and references therein). The first scheme, requiring stronger smoothness properties of the solution trajectory of (1), is based on the modified Euler (Euler-Cauchy) method for solving differential equations and is shown to have the grid error of order O(h 2 ). On the other hand, the latter scheme, based on the Euler method, has the grid error of order O(h) but requires Lipschitz continuity of the solution trajectory only. We provide numerical experiments, comparing the schemes derived and a standard MATLAB function ODE45, for two simple problems arising in mechanics and electronics, respectively. The results from [3] are extended in several directions. Namely, higher-order and inexact schemes are investigated and a weaker (non-strong) metric regularity is also considered.
In the fourth section, we study regularity of mappings associated with the problem of feasibility in control, which is the problem to find a pair of functions x : [0, T] → R n and u : [0, T] → R m such thaṫ x(t) = g(x(t), u(t)) and f (x(t), u(t)) ∈ U ad for a.e. t ∈ [0, T], x(0) = 0, (2) with T, f and g as before and a given closed convex subset U ad of R d . Note that we request (2) to hold for almost every t only instead of for every t in (1) with F ≡ −U ad and x I = 0. The required 'quality' of the functions x(·) and u(·) will be described later in particular statements. We focus on the interplay between the pointwise conditions and their uniform and infinite-dimensional counterparts. We extend several results from [3].
Basic notation. The distance from a point x to a subset A of a metric space (X, ) is d(x, A) = inf y∈A (x, y). The closure and the interior of A is denoted by cl A and int A, respectively. Given sets C, D ⊂ X, the excess of C beyond D is defined by e(C, D) := sup x∈C d(x, D). We use the convention that inf ∅ := +∞ and as we work with non-negative quantities we set sup ∅ := 0. The closed ball centred at a point x ∈ X with a radius r > 0 is denoted by IB r (x). A set A ⊂ X is locally closed at its point x if there is r > 0 such that the set A ∩ IB r (x) is closed. Any singleton set will be identified with its only element, that is, we write a instead of {a}. By F : X ⇒ Y we denote a set-valued mapping between sets X and Y , its graph, domain, and range are the sets gph F : . We write f : X → Y to emphasize that the mapping f is single-valued. The space of all single-valued linear continuous operators acting between Banach spaces X and Y is equipped with the standard operator norm and denoted by L(X, Y). The space R n is equipped with the Euclidean norm, while the Cartesian product of two or more spaces is considered with the box (product) topology. By a.e. we mean almost every in terms of the Lebesgue measure.

Uniform regularity
In our analysis, we employ two key concepts from set-valued analysis called regularity and strong regularity of a set-valued mapping. Let us emphasize that unlike definitions in [4], we prefer not to include the assumption that the mapping under consideration has a locally closed graph in any definition of regularity.
(ii) globally regular if F is regular on X for Y ; (iii) regular atx forȳ ifȳ ∈ F(x) and there are positive constants a, b, and κ such that The infimum of κ > 0 such that the above inequality holds for some a > 0 and b > 0 is the regularity modulus of F atx forȳ and is denoted by reg(F;x |ȳ).
Clearly, if F is regular atx forȳ with a constant κ and neighbourhoods IB a (x) and IB b (ȳ), then F is regular on IB a (x) for IB b (ȳ) with the same constant. On the other hand, when the sets U and V are neighbourhoods of pointsx andȳ, respectively, andȳ ∈ F(x), then regularity of F on U for V implies regularity of F atx forȳ. The constants are the same again but neighbourhoods may differ [ Let μ > 0 be such that κμ < 1 and let κ > κ/ (1 − κμ). Then for every positive α and β such that and for every mapping g : X → Y satisfying the mapping g+G has the following property: for every y, y ∈ IB β (ȳ) and every Proof: We shall imitate the proof of [4,Theorem 5G.3]. First, suppose that G is regular on IB a (x) for IB b (ȳ) with the constant κ. Choose any α and β, and then any g as in the statement. Then Indeed, fix any such a pair (x, y). Then (4) and (3) imply that Fix any two distinct y, y ∈ IB β (ȳ) and any x ∈ (g + G) −1 (y) ∩ IB α (x). Let r := κ y − y . As r ≤ 2κ β, the first inequality in (3) implies that Consider the mapping It suffices to show that there is a fixed point x of Φ in IB r (x), because then x ∈ (g + G) −1 (y ) and the desired distance estimate holds. To obtain such a point x we are going to apply [4,Theorem 5E.2]. The set := gph Φ ∩ (IB r (x) × IB r (x)) is closed. Indeed, pick any sequence (x n , z n ) in converging to a point (x,z) ∈ X × X. Clearly, (x,z) ∈ IB r (x) × IB r (x). The definition of Φ and (6) imply that for each n ∈ N.

Remark 2.4:
Under the strong regularity, the reasoning used at the end of the proof of Theorem 2.3 implies that the function σ , defined therein, is Lipschitz continuous relative to dom σ ⊂ IB β (ȳ) with the constant κ . If, in addition, then dom σ = IB β (ȳ) and consequently g+G is strongly regular on IB 2κ β+α (x) for IB β (ȳ). Note that (7) holds, for example, when (x,ȳ) ∈ gph G.
We also get the following uniformity result. Proof: Let constants γ and δ along with a pair (x, y) be as in the premise. Set U := IB α−γ (x) and V := IB β−δ (y). We have to show that Fix any such a pair (u, v). Pick an As v ∈ (g + G)(u) ∩ V was arbitrary, the proof is finished.
We show now that the regularity at each point of a compact set implies uniform regularity, that is, we can choose the same constant and neighbourhoods for all points in this set. Theorem 2.6: Let (P, ) be a metric space, let (X, · ) and (Y, · ) be Banach spaces, and let be a compact subset of P × X. Consider a set-valued mapping F : X ⇒ Y and a mapping σ : P × X → Y such that Y has a locally closed graph at (x, 0) and is [strongly] regular at x for 0; (ii) for each z = (p, x) ∈ and each μ > 0 there is δ > 0 such that for each v, v ∈ IB δ (x) and each p ∈ IB δ (p) we have (iii) for each x ∈ X the function σ (·, x) is continuous.
Summarizing, for each z = (p, x) ∈ (intIB rz (p) × intIB rz (x)) ∩ the mapping G p is regular at x for 0 with the constant κ z and neighbourhoods IB κ z βz/3 (x) and IB βz/3 (0), that is, the size of neighbourhoods and the constant of regularity are independent of z in a vicinity ofz. From the open cover- Hence the mapping G p is regular at x for 0 with the constant κ and neighbourhoods IB a (x) and IB b (0).
Under the assumption of strong regularity one uses Remark 2.4 (or the strong regularity part of Theorem 5G.3 in [4]).

Remark 2.7:
Note that (ii) in Theorem 2.6 is satisfied, in particular, when σ has a point-based approximation on in the sense of Robinson [6]. Theorem 2.6 yields [5, Lemma 0]. Moreover, given a non-empty subset of a metric space, define the measure of non-compactness of by Then Theorem 2.6 holds provided that χ( ) is strictly smaller than the infimum of the reciprocal values of the regularity moduli of the mappings appearing in (i). This statement is a key element in the proof of the non-smooth versions of Robinson and Lyusternik-Graves theorems, cf. [1, Step 1] and [2, Lemma 12].
Next statement guarantees uniform [strong] regularity along continuous paths.
Then there are positive constants a, b, and κ such that for each t ∈ [0, T] the mapping G t is [strongly] regular at ϕ(t) for ψ(t) with the constant κ and neighbourhoods IB a (ϕ(t)) and IB b (ψ(t)).
Proof: Apply Theorem 2.6 with P : Clearly, we can replace the interval [0, T] by any compact metric space in the above statement.

Path-following for differential generalized equations
and an initial state x I ∈ R n . If it is not clearly indicated otherwise we impose the following: Standing assumptions (SA). Consider the DGE (1) and suppose that f and g are differentiable functions with a locally Lipschitz continuous derivative, and that F has a closed graph. Further, let a pair of functions (x(·),ū(·)) be a solution of (1) such that both of them are differentiable on [0, T] and have a Lipschitz continuous derivative on this interval.
For an integer N > 1, consider the uniform grid with (x 0 , u 0 ) sufficiently close to (x(0),ū(0)). The reason for allowing x 0 = x I is that for a given time interval I := [−T, T], say, one cannot expect thatū(·) is differentiable on the whole of I in general (for example when a geometric constraint represented by the generalized equation is a variational inequality). However,ū(·) can be piece-wise smooth on I and the starting point x 0 can be viewed as a final iterate obtained by a numerical algorithm on the previous subinterval [−T, 0]. In fact, this is the case in our numerical examples. As noted by an anonymous referee the assumptions on the differentiability ofū(·) could be relaxed by employing the averaged modulus of smoothness to obtain the same estimates when the derivative ofū(·) is of bounded variation only; also one can consider more general Runge-Kutta approximations as in [7]. However, we prefer to keep the presentation as clear as possible and use a modification of the classical trapezoidal rule [8] in our analysis.
Ifū(·) is only Lipschitz continuous on [0, T], one can consider the following iteration: Using a similar technique as in the proof of Theorem 3.2 we obtain:

generated by the iteration (26), with the initial point
The above statement is a slight extension of [3, Theorem 5.1]. Next, we discuss two basic examples from engineering, which can be formulated either as a DGE or an ODE with a Lipschitz continuous right-hand side. We compare schemes (12) and (26) with the method ODE45 which is used with the absolute error tolerance 10 −12 to get a reference solution trajectory. All simulations are performed in MATLAB. Example 3.4: Consider a particle of mass m > 0 connected by a rigid, weightless rod of length > 0 to a base by means of a pin joint that can rotate in a plane due to gravity. In addition, the pendulum can have a contact with two walls made of a very flexible material which are at a distance r > 0 from a pin joint. The contact force acting on the mass at time t is denoted by u(t); and ϕ 1 (t) and ϕ 2 (t) denote the angular displacement and the angular velocity at time t, respectively (see Figure 1). The equations of motion of the system are ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩φ with given initial conditions γ 1 , γ 2 ∈ R, a gravitational acceleration g = 9.81 ms −2 , and u(t) = H(ϕ 1 (t)) describing the dependence of the contact force on the angular displacement. We assume that argsinh(ϕ − arcsin (r/ )) for ϕ > arcsin ( /r), argsinh(ϕ + arcsin (r/ )) for ϕ < − arcsin ( /r), 0 o t h e r w i s e .
The corresponding DGE has form ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩φ where ∂ denotes a subdifferential in the sense of convex analysis. The solution for = m := 1, r := sin 1, T := 2, γ 1 = π/3, and γ 2 = 0 is in Figure 2. The grid errors with respect to the solution obtained by ODE45 are in Figure 3. For both the schemes (12) and (26), we use the discretion step h = 10 −5 and e i = 0, i ∈ {0, 1, . . . , N − 1}. Figure 4 involving the four-diodes bridge full-wave rectifier, a resistor with a resistance R > 0, a capacitor with the capacitance C 0 > 0 and an inductor with the inductance L > 0. Denote v C a voltage   across the capacitor, i C a current through the capacitor, i L a current through the inductor, i DF1 , i DF2 , i DR1 , i DR2 currents through the diodes, and v DF1 , v DF2 , v DR1 , v DR2 voltages across the diodes, respectively. Using the Kirchhoff 's laws, this problem can be described as a particular DGE (see [9]) called a differential linear complementarity problem (system) in the form

Example 3.5: Consider a circuit in
where the symbol ⊥ denotes a complementarity relation, and inequalities in R 4 are understood coordinate-wise. From Hence the problem is equivalent to the system of ordinary differential equations, in the forṁ use the discretion step h = 10 −8 and e i = 0, i ∈ {0, 1, . . . , N − 1}. Graphs of solution components are in Figure 5 while grid errors are in Figure 6. We note that the maximal grid error means the biggest error of elements of u or x at the points of the grid.
To conclude this section, let us point out that a similar technique, can be used also in the case of a parametric generalized equation, which is a problem for a fixed function p : [0, T] → R n , find a function z : [0, T] → R n such that where a constant T > 0, a function f : R n → R n and a set-valued mapping F : R n ⇒ R n are given. This problem can be used, for example, for modelling static problems from electronics, that is, when no capacitors and inductors appear in the circuit [10][11][12][13].
For an integer N > 1, define the uniform grid t i := ih, i ∈ {0, 1, . . . , N}, with a step size h := T/N. Given > 0 and points (e i ) N −1 i=0 in IB h 2 (p(t i+1 )), we study a predictor-corrector scheme in the form where z 0 is sufficiently close to the exact solution of (29) at time t: = 0. Uniform regularity along a continuous path was used in [14] to obtain the following extension of the main result from [15].
[strongly] regular atz(t) for p(t). Then there is α > 0 such that for any > 0 there are constants N 0 ∈ N and c > 0 such that for each N > N 0 and each z 0 ∈ IB h 4 (z(t 0 )), where h := T/N, there are [uniquely determined] points (z i ) N i=1 generated by the iteration (30), with the initial point z 0 and arbitrarily chosen points The point e i appearing in (30) can be interpreted as a sufficiently precise prediction at time t i of the (possibly unknown) value of p(t i+1 ). Then we wait until the precise value of p(t i+1 ) is known and compute a correction z i+1 . On the other hand, taking e i := p( exists and is Lipschitz on [0, T] with the constant 2 . Hence the algorithm proposed in [4, Section 6G] is a particular case of (30). Finally, instead of p(t i+1 ) in the latter inclusion of (30) one can take anyẽ i ∈ IB h 4 (p(t i+1 )), that is, the corrector step can be done via an inexact method (which is always the case in practice). Finally, let us note that sufficient conditions (of different type) guaranteeing the existence of a Lipschitz continuous solution z(·) of (29) can be found either in [14,Theorem 6] or [3,Theorem 11].

Uniform regularity and regularity in function spaces
In case that the solution trajectory is not continuous (or even defined) on the whole time interval we can derive the following statement.
with a continuous f : R n × R m → R d having a continuous derivative ∇ u f and F : R m ⇒ R d having a closed graph. Let := ∪ t∈S (x(t),ū(t)) and for each (x, u) ∈ cl define a mapping Then the following statements are equivalent: (i) for each (x, u) ∈ cl the mapping G x,u is [strongly] regular at u for 0; (ii) there are positive constants a, b, and κ such that for each (x, u) ∈ cl the mapping G x,u is [strongly] regular at u for 0 with the constant κ and neighbourhoods IB a (u) and IB b (0); (iii) there are positive constants a, b, and κ such that for each t ∈ S the mapping G t in (13) is [strongly] regular atū(t) for 0 with the constant κ and neighbourhoods IB a (ū(t)) and IB b (0).

Proof:
Assume that (i) holds. Define a (compact) set := cl(∪ t∈S (x(t),ū(t), u(t))) and a (continuous) function σ (x, u, v) if and only if v = u and (x, u) ∈ cl . Theorem 2.6 yields positive constants a, b, and κ such that for each (x, u, u) ∈ the mapping G x,u is [strongly] regular at u for 0 with the constant κ and neighbourhoods IB a (u) and IB b (0). Since (x(t),ū(t),ū(t)) ∈ and G t = Gx (t),ū(t) for each t ∈ S, (iii) is proved.
The above statement is a generalization of [3,Theorem 7], where strong regularity is considered only, because it requests point-wise [strong] regularity on the closure of the range of the solution instead of on the closure of its graph. The functionx(·) can be either an input signal in a parametric generalized equation (29) or a state trajectory of the DGE (1). In the latter case,x(·) is continuous on S = [0, T], so ifū(·) has closed range, then the uniform [strong] regularity of G t in (13) on S is equivalent to its point-wise [strong] regularity on S. We also get the following uniform version of the Lyusternik-Graves and Robinson theorem which implies [3,Theorem 9] under substantially weaker assumptions.

Theorem 4.2:
Let T, S,x(·),ū(·), f, and F be as in Theorem 4.1. Then the mapping G t = f (x(t), ·) + F is [strongly] regular atū(t) for 0 uniformly in t ∈ S if and only if so is the mapping G t in (13).
Proof: Suppose that there are positive constants a, b and κ such that for each t ∈ S the mapping G t in (13) is [strongly] regular atū(t) for 0 with the constant κ and neighbourhoods IB a (ū(t)) and IB b (0). Let β, κ , μ, r, be as in the proof of (iii) ⇒ (ii) in Theorem 4.1. Fix any t ∈ S.
, v ∈ R m . Then g t (ū(t)) = 0 and for any v, v ∈ IB 2κ β+β (ū(t)) we have As in Theorem 4.1 we conclude that the mapping G t = g t + G t is [strongly] regular atū(t) for 0 uniformly in t ∈ S. The converse implication follows in the same way.
Before continuing we set up notions used later. and W := R × P. Given a solution (x(·),ū(·)) ∈ V of (2) we set along with its shifted partial linearization H at (x(·),ū(·)) defined for each (z(·), v(·)) ∈ V by  (i) H is regular at (x(·),ū(·)) for 0; (ii) H is regular at (0, 0) for 0; (iii) K is regular at 0 for 0; (iv) there is a subset S of [0, T] having full Lebesgue measure such that the mapping G t is regular atū(t) for 0 uniformly in t ∈ S; (v) there is a subset S of [0, T] having full Lebesgue measure such that the mapping G t is regular atū(t) for 0 uniformly in t ∈ S; for a.e. t ∈ [0, T]; (vii) there are δ > 0 and r > 0 such that for every w(·) ∈ P with w(·) ∞ < δ there is a pair (z(·), v(·)) ∈ rIB X × rIB U such that
Suppose that (vii) holds. We shall establish (v). Pick β > 0 such thatw β (·) ≡ (β, β, . . . , β) ∈ R d has w β (·) ∞ < δ. Let {w 1 , w 2 , . . . } be a countable dense subset of IB β (0). For any i ∈ N, the function w i (·) ≡ −w i has w i (·) ∞ ≤ w β (·) ∞ < δ, thus there is a subset S i of [0, T] having a full Lebesgue measure along with a pair (z i (·), v i (·)) ∈ rIB X × rIB U such that Without any loss of generality assume that z i (t) ≤ r and v i (t) ≤ r whenever t ∈ S i . Then S := ∩ ∞ i=1 S i has a full Lebesgue measure. Without any loss of generality assume that C(t) ≤ ν andū(t) is defined whenever t ∈ S. Fix any t ∈ S. Define a mapping F t (z, v) :=f (t) + C(t) z + D(t) v − U ad , (z, v) ∈ R n × R m . For every i ∈ N we have w i ∈ F t (rIB R n × rIB R m ). Hence the image of rIB R n × rIB R m under F t (having a closed convex graph) is dense in IB β (0), and consequently applying Robinson-Ursescu theorem [16,Theorem 6.22] we get that F t is regular at (0, 0) for 0 with modulus r/β. In particular, the regularity modulus does not depend on the choice of t ∈ S. Let be the set in Theorem 4.1. Fix any (x, u) ∈ cl . Let Then 0 ∈ F x,u (0, 0) since f is continuous and U ad is closed. Since ∇ x f and ∇ u f are continuous, the uniformity of the regularity moduli of mappings F t and the Lyusternik-Graves theorem imply that F x,u is regular at (0, 0) for 0. Thus the mapping F x,u (z, v) := F x,u (z, v − u), (z, v) ∈ R n × R m , is regular at (0, u) for 0. Since w ∈ F x,u (z, v) if and only if w − ∇ x f (x, u)z ∈ G x,u (v), where G x,u is the mapping in (32) with F ≡ −U ad , we conclude that G x,u is regular at u for 0. Theorem 4.1 implies that (v) holds.
It seems that one can formulate a similar statement when a constant mapping F ≡ −U ad is replaced by a general F : R m → R d with a closed convex graph, which would be a regularity version of [3,Theorem 13]. This is out of the scope of the current work and is a subject for future research.