Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control

Abstract

This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram–Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example.

Appendices

Appendix A: Proof of Theorem 4.1

This section aims at constructing convergent bounds on the set ${\mathbb{E}}_{M}(t,a)$, thereby providing a proof of Theorem 4.1. Recall the definition of the set ${\mathbb{E}}_{M}(t,a)$ in Sect. 4 as

$${\mathbb{E}}_M(t,a) := \bigl\{ x(t,u) - y(t,a) \mid u \in\mathbb{U}_M(a) \bigr\} . $$

The blanket Assumption A2 guarantees that the solution trajectories x and y are well defined, so that a sufficiently large and compact a priori enclosure $\widetilde{\mathbb{E}} \subseteq\mathbb{R}^{n_{x}}$ can always be found such that

$$\forall t \in[0,T] , \quad{\mathbb{E}}_M(t,a) \subseteq\widetilde{ \mathbb{E}} . $$

The focus here is on tightening the a priori enclosure $\widetilde {\mathbb{E}}$, which is crucial for analyzing the convergence of the set ${\mathbb{E}}_{M}(t,a)$. We start by defining the response defect

$$d(t,u,a) = x(t,u) - y(t,a) , $$

which by construction satisfies an ODE of the form

$$\begin{aligned} & \quad\forall t \in[0,T] , \\ & \dot{d}(t,u,a) \overset{(\scriptsize3,9)}{=} f\bigl(x(t,u)\bigr) - f\bigl(y(t,a)\bigr) + G\bigl(x(t,u)\bigr) u(t) - G\bigl(y(t,a)\bigr) \Biggl( \sum _{i=0}^{M} a_i \varPhi_i(t) \Biggr) \\ & \quad\text{with} \quad d(0,u,a) = 0 . \end{aligned}$$

(25)

In order to bound the solution trajectory of (25), consider a Taylor expansion of the term f(x(t,u))−f(y(t,a)) in the form

$$f\bigl(x(t,u)\bigr) - f\bigl(y(t,a)\bigr) = \frac{\partial f(y(t,a))}{\partial x} d(t,u,a) + R_f(t,u,a) d(t,u,a) , $$

where the matrix $R_{f}(t,u,a) \in\mathbb{R}^{n_{x} \times n_{x}}$ denotes a nonlinear remainder term. By the mean-value theorem, R _f is bounded as

$$\begin{aligned} \forall u \in\mathbb{U}_M(a) , \quad\big\Vert R_f(t,u,a) \big\Vert\leq\max_{d',d'' \in\widetilde{\mathbb{E}}} \biggl \Vert \frac{1}{2} \frac{\partial^2 f(y(t,a) + d')}{\partial x^2} d'' \biggr \Vert . \end{aligned}$$

(26)

Similarly, the mean-value theorem can be applied to the term [G(x(t,u))−G(y(t,a))]u(t), giving

$$\bigl[ G\bigl(x(t,u)\bigr) - G\bigl(y(t,a)\bigr) \bigr] u(t) = R_G(t,u,a) d(t,u,a) , $$

where the nonlinear remainder term R _G (t,u,a) is bounded by

$$\begin{aligned} \forall u \in\mathbb{U}_M(a) , \quad\big\Vert R_G(t,u,a)\big \Vert\leq\max_{u' \in\mathbb{F}_u(t)} \max _{d' \in\widetilde{\mathbb{E}}} \biggl \Vert \frac{\partial G( y(t,a) + d')}{\partial x} u' \biggr \Vert . \end{aligned}$$

(27)

In the following, we introduce the shorthand notation R(t,u,a):=R _f (t,u,a)+R _G (t,u,a), so that (25) can be written in the form

$$\begin{aligned} &\quad \forall t \in[0,T] , \\ & \dot{d}(t,u,a) = \frac{\partial f(y(t,a))}{\partial x} d(t,u,a) + R(t,u,a) d(t,u,a) \\ &\phantom{\dot{d}(t,u,a)=}{}+ G \bigl(y(t,a)\bigr) \Biggl[ u(t) - \sum_{i=0}^{M} a_i \varPhi_i(t) \Biggr] \quad\text{with} \ d(0,u,a) = 0 . \end{aligned}$$

(28)

A few remarks are in order regarding the previous ODE (28):

• 1. The Jacobian of f and the function G are evaluated at the point y(t,a). Therefore, these matrices are independent of the control input u.

• 2. The matrix-valued remainder term R(t,u,a) depends on u in general, but standard tools such as interval arithmetics can be applied to construct a bound $\overline{R}(t,a)<\infty$ independent of u such that

  $$\begin{aligned} & \quad \forall t\in[0,T] , \\ & \max_{d',d'' \in\widetilde{\mathbb{E}}} \biggl \Vert \frac {\partial^2 f(y(t,a) + d')}{\partial x^2} \frac{d''}{2} \biggr \Vert _2 + \max_{u' \in \mathbb{F}_u(t)} \max_{d' \in\widetilde{\mathbb{E}}} \biggl \Vert \frac {\partial G( y(t,a) + d')}{\partial x} u' \biggr \Vert _2 \leq\overline{R}(t,a) . \end{aligned}$$

  (29)

  The validity of this bound follows from the inequalities (26) and (27):

  $$\forall u \in\mathbb{U}_M(a) , \quad\bigl \Vert R(t,u,a) \bigr \Vert _2 \leq\overline{R}(t,a) . $$

  However, the tightness of the bound $\overline{R}(t,a)$ depends strongly on the tightness of the a priori enclosure $\widetilde {\mathbb{E}}$, and a tighter $\widetilde{\mathbb{E}}$ will typically lead to a tighter bound on the norm of the remainder term R.

• 3. The right-hand side of the ODE (28) also depends on the control parameterization defect $(u - \sum_{i=0}^{M} a_{i} \varPhi_{i})$. It follows by the orthogonality of the Φ _i s that the first M+1 Gram–Schmidt coefficients of this defect are all equal to zero. Consequently, an enclosure $\mathbb{W}_{M}(a)$ of the control parameterization defect, which is independent of u and convex (Assumption A4), is obtained as:

  $$\begin{aligned} & \quad\forall u \in\mathbb{U}_M(a) , \\ & u - \sum_{i=0}^{M} a_i \varPhi_i \in\mathbb{W}_M(a) := \left \{ w \in L^2[0,T]^{n_u} \left|\begin{array}{l} \forall i \in\{ 0, \ldots, M \}, \forall t \in[0,t], \\ 0 = \langle w , \varPhi_i \rangle_\mu\\ \displaystyle w(t) + \sum_{i=0}^{M} a_i \varPhi_i(t) \in \mathbb{F}_u(t) \end{array} \right. \right \}. \end{aligned}$$

  (30)

Summarizing the above considerations, the main idea is to regard the solution trajectory of the differential equation (28) as a function of the remainder term and of the control parameterization defect rather than as a function of the control function u. In order to formalize this change of variable, let e(t,Δ,w) denote the solution of the ODE

$$\begin{aligned} \dot{e}(t,\Delta,w) =& \bigl[ A(t) + \Delta(t) \bigr] e(t,\Delta,w) + B(t) w(t) , \quad\text{with}\ e(0,\Delta,w) = 0, \end{aligned}$$

(31)

for any functions $w \in L^{2}[0,T]^{n_{u}}$ and $\Delta\in L^{2}[0,T]^{n_{x} \times n_{x}}$, where we have introduced the shorthand notations $A(t) := \frac {\partial f(y(t,a))}{\partial x}$ and B(t):=G(y(t,a)).

Proposition A.1

Let $\overline{R}(\cdot,a)$ be a remainder bound satisfying (29), and let $\mathbb{W}_{M}(a)$ be defined by (30). Then,

$$\begin{aligned} &\quad\forall(t,a) \in[0,T]\times\mathbb{D}_M , \\ &{\mathbb{E}}_M(t,a) \subseteq \left \{ e(t,\Delta,w) \left| \begin{array}{l} w \in\mathbb{W}_M(a) \\ \forall\tau\in[0,T] \quad\Vert\Delta(\tau) \Vert_2 \leq \overline{R}(\tau,a) \end{array} \right.\right \} , \end{aligned}$$

where e(⋅,Δ,w) denotes the solution of (31).

Proof

From the definition of the defect function d we have

$${\mathbb{E}}_M(t,a) = \bigl\{ d(t,u,a) \mid u \in\mathbb{U}_M(a) \bigr\}. $$

Moreover, the function e is defined such that

$$d(t,u,a) = e \Biggl(t, R(\cdot,u,a), u - \sum_{i=0}^{M} a_i \varPhi_i \Biggr) . $$

Therefore, the result of the proposition follows by applying the change of variables

$$R(\cdot,u,a) \rightarrow\Delta\quad\text{and} \quad u - \sum _{i=0}^{M} a_i \varPhi_i \rightarrow w $$

and using that $\overline{R}(\cdot,a)$ and $\mathbb{W}_{M}(a)$ are bounds on the norm of the remainder term and on the control parameterization defect, respectively. □

In order to understand the motivation behind Proposition A.1, it is helpful to interpret the differential equation (31) as a low-pass filter with uncertain but bounded gain Δ, which would filter high-frequency modes of the “noise” w. This interpretation is useful, as the control parameterization defect $w = u - \sum_{i=0}^{M} a_{i} \varPhi_{i}$ is bounded by the set W _M (a). For example, if Φ ₀,Φ ₁,… denote the basis functions in a standard trigonometric Fourier expansion, the set $\mathbb{W}_{M}(a)$ contains for large M only highly oscillatory functions, as the first M+1 Fourier coefficients of the control parameterization are zero in this case. The differential equation (31) can be expected to filter out these highly oscillating modes such that e(t,Δ,w)≈0. Having this interpretation in mind, intuition suggests that we should be able to compute tight bounds on the function e(t,Δ,w), which converge for M→∞, even when the uncertain gain Δ is not known exactly.

The aim of the following considerations is to formalize this intuition by translating it into a rigorous algorithm that computes convergent enclosures of the sets ${\mathbb{E}}_{M}(\cdot,a)$ on [0,T]. To do so, the function e(t,Δ,w) is decomposed into the sum of two functions e _L(t,w) and e _N(t,Δ,v) such that

$$e(t,\Delta,w) = e_\mathrm{L}(t,w) + e_\mathrm{N}\bigl(t, \Delta,e_\mathrm{L}(\cdot,w)\bigr). $$

Specifically, e _L(⋅,w) and e _N(⋅,Δ,v) are the solutions of the following auxiliary ODEs:

$$\begin{aligned} & \quad\forall t\in[0,T] , \\ & \dot{e}_\mathrm{L}(t,w) = A(t) e_\mathrm{L}(t,w) + B(t) w(t) \quad\text{with}\ \ e_\mathrm{L}(0,w) = 0 , \end{aligned}$$

(32)

$$\begin{aligned} & \dot{e}_\mathrm{N}(t, \Delta,v) = \bigl(A(t) + \Delta(t) \bigr) e_\mathrm{N}(t,\Delta ,v) + \Delta(t) v(t) \quad\text{with} \ \ e_\mathrm{N}(t,\Delta,v) = 0 . \end{aligned}$$

(33)

Note that the output $e_{\rm L}$ of the first ODE (32) becomes an input in the second ODE (33). Moreover, we define the sets

$$\begin{aligned} {\mathbb{E}}_M^\mathrm{L}(t,a) :=& \bigl\{ e_\mathrm{L}(t,w) \mid w \in\mathbb{W}_M(a) \bigr\} \quad\text{and} \\ {\mathbb{E}}_M^\mathrm{N}(t,a) :=& \left \{ e_\mathrm{N}(t,\Delta,v) \left|\forall\tau\in[0,t] , \quad \begin{array}{l} v(\tau) \in{\mathbb{E}}_M^\mathrm{L}(\tau,a) \\ \Vert\Delta(\tau) \Vert_2 \leq{\overline{R}}(\tau,a) \end{array} \right. \right \} . \end{aligned}$$

The following proposition is a direct consequence of Proposition A.1 and of the foregoing decomposition.

Proposition A.2

The image set ${\mathbb{E}}_{M}(t,a)$ is contained in the Minkowski sum of the sets ${\mathbb{E}}_{M}^{\mathrm{L}}(t,a)$ and ${\mathbb{E}}_{M}^{\mathrm{N}}(t,a)$,

$$ \forall(t,a) \in[0,T]\times\mathbb{D}_M , \quad{\mathbb{E}}_M(t,a) \subseteq{\mathbb{E}}_M^\mathrm{L}(t,a) \oplus{\mathbb{E}}_M^\mathrm{N}(t,a) . $$

(34)

The following lemma establishes that, in order to find a convergent bound on the diameter of the set ${\mathbb{E}}_{M}(t,a)$, it is sufficient to find a convergent bound on the diameter of the set ${\mathbb{E}}_{M}^{\mathrm{L}}(t,a)$.

Lemma A.1

There exists a constant ℓ(t)<∞ such that

$$\forall t \in[0,T] , \quad\operatorname{diam}\bigl({\mathbb{E}}_M(t,a)\bigr) \leq\ell(t) \max_{\tau\in[0,T]} \operatorname{diam}\bigl({\mathbb{E}}_M^\mathrm{L}( \tau,a)\bigr) . $$

Proof

Since the uncertain gain Δ is bounded, there exist constants ℓ ₁ and ℓ ₂ such that

$$\bigl \Vert A(t) + \Delta(t) \bigr \Vert _2 \leq \ell_1 \quad\text{and} \quad\bigl \Vert \Delta(t)v(t) \bigr \Vert _2 \leq\ell_2 \frac{\operatorname{diam}({\mathbb{E}}_M^\mathrm{L}(t,a))}{2} $$

for all v with $v(t) \in{\mathbb{E}}_{M}^{\mathrm{L}}(t,a)$, all Δ with $\Vert\Delta(t) \Vert_{2} \leq{\overline{R}}(t,a)$, and all t∈[0,T]. Applying Gronwall’s lemma for bounding the norm of the solution of the differential equation (33) yields the following bound on the diameter of ${\mathbb{E}}_{M}^{\mathrm{N}}(t,a)$:

$$\operatorname{diam}\bigl({\mathbb{E}}_M^\mathrm{N}(t,a)\bigr) \leq\exp( \ell_1 t ) \ell_2 \max_{\tau\in[0,T]} \operatorname{diam}\bigl({ \mathbb{E}}_M^\mathrm{L}(\tau,a)\bigr) . $$

Then from Proposition A.2 it follows that

$$\begin{aligned} \operatorname{diam}\bigl({\mathbb{E}}_M(t,a)\bigr) \leq& \operatorname {diam}\bigl({\mathbb{E}}_M^\mathrm{L}(t,a)\bigr) + \operatorname{diam}\bigl({\mathbb{E}}_M^\mathrm{N}(t,a)\bigr) \\ \leq& \bigl[ 1 + \ell_2 \exp( \ell_1 t ) \bigr] \max _{\tau\in[0,T]} \operatorname{diam}\bigl({\mathbb{E}}_M^\mathrm{L}( \tau ,a)\bigr) , \end{aligned}$$

(35)

and the result follows by defining ℓ(t):=1+ℓ ₂exp(ℓ ₁ t). □

In the remainder of this appendix, the focus is on bounding the set ${\mathbb{E}}_{M}^{\mathrm{L}}(t,a)$. We start by noting that the image sets ${\mathbb{E}}_{M}^{\mathrm{L}}(t,a)$ are both compact and convex. The compactness of ${\mathbb{E}}_{M}^{\mathrm{L}}(t,a)$ follows from Filippov’s theorem [62]. Moreover, the convexity of ${\mathbb{E}}_{M}^{\mathrm{L}}(t,a)$ follows from the fact that linear transformations of convex sets are convex, recalling that the set $\mathbb{W}_{M}(a)$ is convex and that the functional e _L(t,⋅) is linear since the ODE (32) is itself linear; see also [71]. Because any compact and convex set is uniquely characterized by its support function [72], we define the support function $V(t,\cdot): \mathbb{R}^{n_{x}} \to\mathbb{R}$ associated with the image set ${\mathbb{E}}_{M}^{\mathrm{L}}(t,a)$ by

$$ \forall c \in\mathbb{R}^{n_x} , \quad V(t,c) := \max _{w \in \mathbb{W}_M(a)} c^\mathsf {T}e_\mathrm{L}(t,w) . $$

(36)

It is not hard to see from (36) that

$$\sup_{c} \frac{V(t,c)}{\Vert c \Vert_2} \geq\frac{1}{2} \operatorname{diam}\bigl({\mathbb{E}}_M^\mathrm{L} ( t,a ) \bigr) , $$

thereby providing a means for bounding the diameter of ${\mathbb{E}}_{M}^{\mathrm{L}} ( t,a )$. In expanded form, V(t,c) reads

$$\begin{aligned} V(t,c) :=& \displaystyle\max_{w\in L^2[0,T]^{n_u}}\ c^\mathsf {T}\int _0^t Z(t,\tau) B(\tau) w(\tau) \,\mathrm{d}\tau \\ &\text{s.t.} \quad \left \{ \begin{array}{l} \forall i \in\{ 0, \ldots, M \}\quad \forall t \in[0,t] ,\\ 0 = \langle w , \varPhi_i \rangle_\mu, \\ \displaystyle w(t) + \sum_{i=0}^{M} a_i \varPhi_i(t) \in \mathbb{F}_u(t), \end{array} \right . \end{aligned}$$

(37)

with $Z: [0,T] \times[0,T] \to\mathbb{R}^{n_{x} \times n_{x}}$ the fundamental solution of the parametric linear ODE (32), which satisfies

$$\forall\tau,t \in[0,T] , \quad\frac{\partial}{\partial t} Z(t,\tau) = A(t) Z(t,\tau) \quad\text{with} \ Z(\tau,\tau) = I. $$

Since the sets $\mathbb{F}_{u}(t)$ are compact, one can always scale the dynamic system, and we shall therefore restrict the analysis to the L ²-integrable functions w with $w(t)+\sum_{i=0}^{M} a_{i} \varPhi _{i}(t)\in\mathbb{F}_{u}(t)$ for all t∈[0,T] and such that

$$\Vert w \Vert_{\mu,2} := \int_0^T w(t)^\mathsf {T}w(t) \mu(t) \,\mathrm{d}t \leq1 . $$

Let the functions $H_{t}: [0,T] \to\mathbb{R}^{n_{x} \times n_{x}}$ and $\theta: [0,T] \times[0,T] \to\mathbb{R}$ be defined by

$$H_{t}(\tau) := \frac{1}{\mu(\tau)} Z(t,\tau) B(\tau) \theta (t,\tau) \quad\text{and} \quad\theta(t,\tau) := \left \{ \begin{array}{l@{\quad}l} 1 & \text{if $\tau\leq t$} , \\ 0 & \text{otherwise} , \end{array} \right . $$

for all t,τ∈[0,T]. The objective function in (38) can be written in the form

$$\begin{aligned} c^\mathsf {T}\int_0^t Z(t,\tau) B(\tau) w( \tau) \,\mathrm{d}\tau =& c^\mathsf {T}\int_0^T \frac{1}{\mu(\tau)} Z(t,\tau) B(\tau) \theta(t,\tau) w(\tau) \mu(\tau) \,\mathrm{d} \tau\\ =& \bigl\langle c^\mathsf {T}H_{t} , w \bigr \rangle_\mu, \end{aligned}$$

and an upper bound on the support function V(t,c) is given by

$$\begin{aligned} V(t,c) \leq& \max_{w} \bigl\langle c^\mathsf {T}H_{t} , w \bigr\rangle_\mu\quad \text{s.t.} \quad \left \{ \begin{array}{l@{\quad}l} 0 = \langle w , \varPhi_i \rangle_\mu& \forall i \in\{ 0, \ldots, M \} , \\ \Vert w \Vert_{\mu,2} \leq 1. \end{array} \right . \end{aligned}$$

(38)

Associating with each equality constraint 0=〈w,Φ _i 〉 _μ in (38) the scaled multipliers $c^{\mathsf {T}}D_{i} \in\mathbb{R}^{n_{u}}$, the dual of the convex problem (38) gives

$$\begin{aligned} V(t,c) \leq& \inf_{D} \max_w \bigl \langle c^\mathsf {T}H_{t} , w \bigr\rangle_\mu- \sum _{i=0}^M c^\mathsf {T}D_i \langle\varPhi_i , w \rangle_\mu\quad\text{s.t.} \quad \Vert w \Vert_{\mu,2} \leq1 \\ = & \inf_{D} \Biggl \Vert c^\mathsf {T}\Biggl( H_{t} - \sum_{i=0}^M D_i \varPhi_i \Biggr) \Biggr \Vert _{\mu,2} = \Biggl \Vert c^\mathsf {T}\Biggl( H_{t} - \sum _{i=0}^M \frac{\langle H_{t} , \varPhi_i \rangle_\mu}{\sigma_i} \varPhi_i \Biggr) \Biggr \Vert _{\mu,2} , \end{aligned}$$

(39)

where the tight version of the Cauchy–Schwarz inequality for L ₂-scalar products has been used in the last equality. At this point, it becomes clear that the derived bound on the support function V(t,c) depends crucially on how accurately the function H _t can be approximated with the function $\sum_{i=0}^{M} D_{i} \varPhi_{i}$. This observation is formalized in the following theorem.

Theorem A.1

Let Assumption 4.1 and blanket Assumptions A2 and A4 be satisfied. Then, there exist constants $C_{E}^{L,0} \in\mathbb{R}$ and $C_{E}^{L,1} \in\mathbb{R}_{++}$ such that the condition

$$\forall(t,a) \in[0,T]\times\mathbb{D}_M , \quad\log\bigl( \operatorname{diam}\bigl({ \mathbb{E}}_M^\mathrm{L} ( t,a ) \bigr) \bigr) \leq C_E^{L,0} - C_E^{L,1} M $$

is satisfied for all $M \in\mathbb{N}$.

Proof

Since the function H _t is piecewise smooth and, by Assumption 4.1, there exist a sequence $D_{0},D_{1},\ldots\in\mathbb{R}^{n_{x} \times n_{u}}$ and constants $C_{H}^{0} \in\mathbb{R}$ and $C_{H}^{1} \in \mathbb{R}_{++}$ such that

$$\begin{aligned} \log\Biggl( \Biggl \Vert H_{t}(\tau) - \sum _{i=0}^M D_i \varPhi_i(\tau) \Biggr \Vert \Biggr) \leq& C_H^0 - C_H^1 M \end{aligned}$$

(40)

for almost all τ∈[0,T] and all $M \in\mathbb{N}$. By combining (38) and (40), it follows that there exists a constant $C_{\mu}\in\mathbb{R}_{++}$ such that

$$\begin{aligned} \log\bigl( \operatorname{diam}\bigl({\mathbb{E}}_M^\mathrm{L} ( t,a ) \bigr) \bigr) \leq\log\biggl( 2 \sup_{c} \frac{V(t,c)}{\Vert c \Vert_2} \biggr) \leq& 2 C_\mu\bigl[ C_H^0 - C_H^1 M \bigr] \end{aligned}$$

(41)

for almost all τ∈[0,T] and all $M \in\mathbb{N}$. In the last step, we have used the Lebesgue dominated convergence theorem (or alternatively Fatou’s lemma), which guarantees that the Lebesgue zero measure of points τ∈[0,T] at which inequality (40) may be violated does not contribute to the bound on the L ₂-norm of the function $H_{t} - \sum_{i=0}^{M} D_{i} \varPhi_{i}$. The statement of the theorem follows by taking $C_{E}^{L,0} := 2 C_{\mu}C_{H}^{0}$ and $C_{E}^{L,1} := 2 C_{\mu}C_{H}^{1}$. □

Finally, a proof of Theorem 4.1 is obtained by combining the results in Theorem A.1 and Lemma A.1.

Appendix B: Computation of Convergent Enclosures $\overline{\mathbb{E}}_{M}(t,\mathbb{A})$

Since the proof in Appendix A is essentially constructive, a line-by-line transcription of this proof into a numerical bounding procedure that constructs an enclosure $\overline {\mathbb{E}}_{M}(t,\mathbb{A})$ satisfying Assumption 5.2 is in principle possible, assuming that suitable interval tools are available. For the implementation in this paper, we have used the bounding and relaxation techniques available through the library MC++ [50, 58], and we have refined the procedure based on the following observations:

• 1. From inequality (39) we know that the set ${\mathbb{E}}_{M}^{\mathrm{L}}(t,a)$ is enclosed by an ellipsoid of the form $\mathcal{E}(Q(t,a))$ with

  $$ Q(t,a) := \int_0^T \Biggl( H_{t}( \tau) - \sum_{i=0}^M \frac {\langle H_{t}, \varPhi_i \rangle_\mu}{\sigma_i} \varPhi_i(\tau) \Biggr) \Biggl( H_{t}(\tau) - \sum _{i=0}^M \frac{\langle H_{t}, \varPhi_i \rangle_\mu }{\sigma_i} \varPhi_i(\tau) \Biggr)^\mathsf {T}\,\mathrm{d}\tau. $$

  By using standard tools from interval analysis for differential equations, a convergent bounding matrix $\mathbb{Q}(t,\mathbb{A}) \in \mathbb{R}^{n_{x} \times n_{x}}$ can be constructed such that $Q(t,a) \preceq \mathbb{Q}(t,\mathbb{A})$ for all $a \in\mathbb{A}$ and all t∈[0,T]. In particular, if there exists a constant C _Q such that

  $$\bigl \Vert \mathbb{Q}(t,\mathbb{A}) - \mathbb{Q}\bigl(t,\{ \operatorname {mid}(\mathbb{A}) \} \bigr) \bigr \Vert \leq C_Q \operatorname{diam}(\mathbb{A}) , $$

  then the enclosure $\overline{\mathbb{E}}_{M}^{\mathrm{L}}(t,\mathbb{A}) := \mathcal{E}( \mathbb{Q}(t,\mathbb{A}) )$ will satisfy the convergence conditions

  $$\begin{aligned} \log\bigl(\operatorname{diam}\bigl(\overline {\mathbb{E}}_M^\text{L}\bigl(t, \{ \operatorname {mid}(\mathbb{A}) \} \bigr)\bigr) \bigr ) \leq& C_{\mathbb{E}}^{\text{L},0} - C_{\mathbb{E}}^{\text {L},1} M , \end{aligned}$$

  (42)
  $$\begin{aligned} \operatorname{diam}\bigl(\overline{\mathbb{E}}_M^\text{L} ( t, \mathbb{A} ) \bigr) - \operatorname{diam}\bigl(\overline {\mathbb{E}}_M^\text{L}\bigl(t, \{ \operatorname {mid}(\mathbb{A}) \}\bigr)\bigr) \leq& C_{\mathbb{E}}^{\text{L},2} \operatorname{diam}(\mathbb{A}) , \end{aligned}$$

  (43)

  for all compact sets $\mathbb{A}$, for all $M \in\mathbb{N}$, and for some constants $C_{\mathbb{E}}^{\text{L},0} \in\mathbb{R}$, $C_{\mathbb{E}}^{\text{L},1} \in\mathbb{R}_{++}$, and $C_{\mathbb{E}}^{\text{L},2} \in\mathbb{R}_{+}$.

• 2. The proof of Lemma A.1 is based on Gronwall’s lemma, which is known to provide mathematically valid, yet typically conservative, bounds for practical purposes. Instead, our implementation considers a modified version of state-of-the-art reachable set enclosure algorithms for bounding the solution of the differential equation (33), which prove to be much less conservative than with Gronwall’s lemma.