A continuous linear dynamical system (CDS) is a system whose evolution is governed by a differential equation of the form , where is a matrix with real entries. CDSs are ubiquitous in mathematics, physics, and engineering, they have been extensively studied as they describe the evolution of many types of systems (or abstractions thereof) over time. More recently, CDSs have become central in the study of cyber-physical systems; a recent authoritative textbook on the matter is [alur2015principles].
In the study of CDSs, particularly from the perspective of control theory, a fundamental problem is reachability – namely whether the orbit , intersects a given set . For example, when describes the state of an autonomous car (i.e., its location, velocity, etc.) may describe situations where the car is not able to stop in time to respond to a hazard.
When is a singleton set, reachability is decidable [hainry, Theorem 2]. However, already when is a half-space, reachability is not known to be decidable. This open problem is known in the literature as the continuous Skolem problem. Some partial positive results were given in [Bell10] and [chonev2016skolem]. The continuous Skolem problem is related to notoriously difficult problems in the theory of Diophantine approximations: a procedure for the continuous Skolem problem would yield one for computing the Diophantine-approximation types (a central quantity in this field) of all real algebraic numbers [chonev2016skolem].
One approach, in lieu of a direct (non) reachability algorithm, is to find a set that separates the orbit from . In order for this scheme to be useful, structural restrictions are placed on to make it easy to verify that contains the orbit, and that it is disjoint from (indeed, if we give up either requirement, we can use as either the orbit itself, or , neither of which makes the problem any easier).
Sensible candidates for such structured sets are inductive invariants. These are sets that, in addition to containing the orbit, are also invariant under the dynamics of the system. If is invariant, proving that the orbit is contained in amounts to proving that the starting point belongs to , which is typically easy. Further restricting the class of sets under consideration (e.g., polyhedra, semi-algebraic sets, etc.), testing whether intersects becomes, likewise, easy.
In [ACOW20, ACOW18], a non-empty intersection of the authors studied o-minimal invariants for discrete linear dynamical systems. There it is proved that when is a semi-algebraic set, the question of whether there exists a o-minimal invariant disjoint from is decidable. Moreover, on the positive instances a semi-algebraic invariant exists and it can be constructed effectively. The present paper follows along similar ideas, although continuous linear dynamical systems reveal new difficulties.
We consider the following problem: given a CDS by means of a matrix with rational entries, an initial point , and a semi-algebraic set , we wish to decide whether there exists a semi-algebraic invariant that is disjoint from . Assuming Schanuel’s conjecture (a unifying conjecture in transcendental number theory), we prove that this problem is decidable and for the positive instances, the invariant can be effectively constructed.
Unconditionally, we can decide a slightly weaker problem, namely the question of whether there exists a semi-algebraic set disjoint from that is: (1) invariant under the dynamics of the system, and (2) contains all but a finite part of the orbit. Likewise, when the answer is positive, the set can be effectively constructed. Such an invariant serves as a certificate that the orbit does not enter a semi-algebraic set infinitely often. The latter is a very difficult problem even when the target set is a half-space [chonev2016recurrent].
As mentioned earlier, for discrete linear dynamical systems the question of whether there exists a semi-algebraic invariant that contains the whole orbit is decidable [ACOW20, ACOW18]. We provide an explanation of why the analogous result for continuous systems is not easy to achieve, by way of a reduction from a difficult problem that highlights the complications of continuous systems. The problem asks whether a given exponential polynomial of the form
has zeros in a bounded interval, where are real algebraic numbers. Deciding whether has zeros in a bounded region seems to be difficult because all the zeros have to be transcendental (a consequence of Hermite-Lindemann Theorem), and they can be tangential, i.e. never changes its sign, yet it has a zero.
Invariant synthesis is a central technique for establishing safety properties of hybrid systems. It has long been known how to compute a strongest algebraic invariant [Rodriguez-CarbonellT05] (i.e., a smallest algebraic set that contains the collection of reachable states) for an arbitrary CDS. Here an algebraic invariant is one that is specified by a conjunction of polynomial equalities. If one moves to the more expressive setting of semi-algebraic invariants, which allow inequalities, then there is typically no longer a strongest (or smallest) invariant, but one can still ask to decide the existence of an invariant that avoids a given target set of configurations. This is the problem that is addressed in the present paper.
Partial positive results are known, for example when strong restrictions on the matrix
are imposed, such as when all the eigenvalues are real and rational, or purely imaginary with rational imaginary part[LafferrierePY01].
A popular approach in previous work has been to seek invariants that match a given syntactic template, which allows to reduce invariant synthesis to constraint solving [GulwaniT08, SturmT11, LiuZZ11]. While this technique can be applied to much richer classes of systems than those considered here (e.g., with discrete control modes and non-linear differential equations), it does not appear to offer a way to decide the existence of arbitrary semi-algebraic invariants. An alternative to the template approach for invariant generation involves obtaining candidate invariants from semi-algebraic abstractions of a system [SogokonGJP16]. Another active area of current research lies in developing powerful techniques to check whether a given semi-algebraic set is actually an invariant [GhorbalSP17, LiuZZ11].
Other avenues for analysing dynamical systems in the literature include non-inductive invariants, such as bisimulations [broucke2002reachability], forward/backward reach-set computation [anai2001reach] and methods for proving directly liveness properties [sogokon2015direct]. The latter depends on constructing staging sets, which are essentially semi-algebraic invariants.
Often, questions about dynamical systems, can be reduced to deciding whether a sentence belongs to the elementary theory of an appropriate expansion of the ordered field of real numbers. While the latter is usually undecidable, there are partial positive results, namely the quasi-decidability in bounded domains, see [franek2016quasi] and the references therein. This can be used to reason about the dynamics of a system in a bounded time interval, under the assumption that it does not tangentially approach the set that we want to avoid. However, it seems unlikely that such results can be easily exploited in the problems considered here.
The rest of the paper is organised as follows. In Section 2, we give the necessary definitions and terminology. In LABEL:sec:_orbitcones, we define cones which are over-approximations of the orbit, and prove that they are in a certain sense canonical. The positive results assuming Schanuel’s conjecture are subsequently given in this section. Section 4 is devoted to the effective construction of the semi-algebraic invariants which allows us to state and prove the unconditional positive results. In Section 5, we give the aforementioned reduction, from finding zeros of exponential polynomials.
A continuous-time linear dynamical system is a pair
where and . The system evolves in time according the function which is the unique solution to the differential equation with . Explicitly this solution can be written as:
The orbit of from time is the set . An invariant for from time is a set that contains and is stable under applications of , i.e., for every . Note that an invariant from time contains . Given a set (referred hence as an error set), we say that the invariant avoids if the two sets are disjoint.
We denote by the structure . This is the ordered field of real numbers with constants and . A sentence in the corresponding first-order language is a quantified Boolean combination of atomic propositions of the form , where is a polynomial with integer coefficients and are variables. In addition to , we also consider its following expansions:
, obtained by expanding with the real exponentiation function .
, obtained by expanding with the restricted elementary functions, namely , , and .
, obtained by expanding with the restricted elementary functions.
Tarski famously showed that the first-order theory of admits quantifier elimination, moreover the elimination is effective and therefore the theory is decidable [tarski1951decision, Theorem 37].
It is an open question whether the theory of the reals with exponentiation () is decidable; however decidability was established subject to Schanuel’s conjecture by MacIntyre and Wilkie [MacintyreWilkie1996, Theorem 1.1]. MacIntyre and Wilkie further showed in [MacintyreWilkie1996, Section 5] that decidability of the theory of implies a weak form of Schanuel’s conjecture.
Similarly, it is an open question whether and are decidable, but they are also known to be decidable subject to Schanuel’s conjecture [macintyre2016turing, Theorem 3.1]111More precisely, the decidability of requires Schanuel’s conjecture over , whereas that of requires it over ..
Let be an expansion of the structure . A set is definable in if there exists a formula in with free variables such that . For , the ordered field of real numbers, -definable sets are known as semi-algebraic sets.
There is a natural first-order interpretation of the field of complex numbers in the field of real numbers . We shall say that a set is -definable if the image of under this interpretation is -definable.
A totally ordered structure is said to be o-minimal if every definable subset of is a finite union of intervals. Tarski’s result on quantifier elimination implies that is o-minimal. The o-minimality of and is shown in [Wilkie96], and the o-minimality of and is due to [van1994elementary, van1996geometric].
A semi-algebraic invariant is one that is definable in . An o-minimal invariant is one that is definable in an o-minimal expansion of .
3 Orbit Cones
In this section we define orbit cones, an object that plays a central role in the subsequent results. They can be thought of as over-approximations of the orbit that has certain desirable properties, and moreover it is canonical in the sense that any other invariant must contain a cone.
3.1 Jordan Normal Form
Let be a continuous linear dynamical system. The exponential of a square matrix is defined by its formal power series as
Let be the eigenvalues of , and recall that when , all the eigenvalues are algebraic. We can write in Jordan Normal Form as where
is an invertible matrix with algebraic entries, andis a block-diagonal matrix where each block is a Jordan block that corresponds to eigenvalue , and it has the form
From the power series, we can write . Further, . For each , write , where is the diagonal matrix and is the matrix ; where is the -th diagonal matrix, with other entries zero.
The matrices and commute, since the former is a diagonal matrix. A fundamental property of matrix exponentiation is that if matrices commute, then . Thus, we have
where by we mean the diagonal matrix that has the entry written times, the entry written times and so on. It will always be clear from the context whether we repeat the entries because of their multiplicity or not.
Matrices are nilpotent, so its power series expansion is a finite sum, i.e. a polynomial in . More precisely, one can verify that:
Set . From the equation above, the entries of are polynomials in with rational coefficients.
Write the eigenvalues as , so that
We have in this manner decomposed the orbit
into an exponential , a rotation , and a simple polynomial matrices that commute with one another. Having the orbit in such a form will facilitate the analysis done in the sequel.
3.2 Cones as Canonical Invariants
In a certain sense, the rotation matrix is the most complicated, because of it, the orbit is not even definable in . The purpose of cones is to abstract away this matrix by a much simpler subgroup of the complex torus
To this end, consider the group of additive relations among the frequencies :
The subgroup of the torus of interest, respects the additive relations as follows:
Its desirable properties are summarised in the following proposition: For algebraic numbers ,
diagonals of form a dense subset of .
Being an Abelian subgroup of , has a finite basis, moreover this basis can be computed because of effective bounds, [Mas88, Section 3]. To check that belongs to , it suffices to check that for in the finite basis. This forms a finite number of equations, therefore
is semi-algebraic. The fact that this is a subset of vectors of complex numbers is not problematic in this case because of the simple first-order interpretation in the theory of reals, seeSection 2.
The second statement of the proposition is a consequence of Kronecker’s theorem on inhomogeneous simultaneous Diophantine approximations, see [cassels1965introduction, Page 53, Theorem 4]. The proof of a slightly stronger statement can also be found in [chonev2016recurrent, Lemma 4]. Examples can be found where the set of diagonals of is a strict subset of .
The orbit cone can now be defined by replacing the rotations with the subgroup of the torus. As it turns out, for our purposes this approximation is not too rough.
The orbit cone from is
We prove that the cone is an inductive invariant and also a subset of .
For all , .
Fix and , and consider the point
then we can write as
The matrix is equal to for some . Otherwise said, the vector belongs to . Indeed this is the case because for any we have
Finally, by induction on the dimension one can verify that .
The fact that cones are subsets of comes as a corollary of the following proposition which is proved in Appendix A.  Let as above, and let for , with dimensions compatible to the Jordan blocks of , and such that for every , if , then . Then has real entries.
The matrix can be written as where the matrices satisfy the conditions of Section 3.2, hence the following corollary. For all we have .
It is surprising that, already, the cones are a complete characterisation of o-minimal inductive invariants in the following sense.  Let be an o-minimal invariant that contains the orbit from some time , then there exists such that:
Conceptually, the proof follows along the lines of its analogue in [ACOW18]. There are a few differences, namely that the entries of the matrix in [ACOW18] are assumed to be algebraic, while this is not true for the entries of .
We define rays of the cone, which are subsets where is fixed. Then we prove that for every ray, all but a finite part of it, is contained in the invariant. This is done by contradiction: if a ray is not contained in the invariant, a whole dense subset of the cone can be shown not to be contained in the invariant, leading to a contradiction, since the invariant is assumed to contain the orbit. We achieve this using some results on the topology of o-minimal sets.
The complete proof deferred to Appendix B.
Another desirable property of cones is that they are -definable. Also, one can observe that for every , the set is definable in (as we only need bounded restrictions of and to capture e.g. up to time ). As an immediate corollary of Section 3.2, we have the following theorems. Let be a CDS. For every , the set is an invariant that contains the whole orbit of . Moreover, this invariant is definable in (and in particular is o-minimal). Let be a CDS and let be an error set. There exists an o-minimal invariant that contains the orbit and is disjoint from if and only if there exists such that is such an invariant.
Section 3.2 now allows us to provide an algorithm for deciding the existence of an invariant, subject to Schanuel’s conjecture: Assuming Schanuel’s conjecture, given a CDS and an definable error set , it is decidable whether there exists an o-minimal invariant for that avoids . Moreover, if such an invariant exists, we can compute a representation of it.
By Section 3.2, there exists an o-minimal invariant that avoids if and only if there exists some such that is such an invariant. Thus, the problem reduces to deciding the truth value of the following sentence:
The theory of is decidable subject to Schanuel’s conjecture, and therefore we can decide the existence of an invariant. Moreover, if an invariant exists, we can compute a representation of it by iterating over increasing values of , until we find a value for which the sentence is true.
4 Semi-algebraic Error Sets and Fat Trajectory Cones
In this section, we restrict attention to semi-algebraic invariants and semi-algebraic error sets, in order to regain unconditional decidability.
Substitute in the definition of the cone to get:
Written this way, observe that , which is almost semi-algebraic, apart from the fact that the exponents need not be rational.
4.1 Unconditional Decidability
We give the final, yet crucial property of the cones. When the error set is semi-algebraic, it is possible to decide, unconditionally, whether there exists some cone that avoids the error set. Moreover the proof is constructive, it will produce the cone for which this property holds. For a semi-algebraic error set , it is (unconditionally) decidable whether there exists such that . Moreover, such a can be computed.
Define the set
The set can be seen to be semi-algebraic and thus is expressed by a quantifier-free formula that is a finite disjunction of formulas of the form , where each is a polynomial with integer coefficients, over variables of the entries of the matrix , and. Define the matrix
and notice that if and only if for every . Thus, it is enough to decide whether there exists such that for every , at least one of the disjuncts is satisfied.
Since are polynomials in entries of the form and , there is an effective bound such that for all , none of the values change sign for any . Hence we only need to decide whether there exists some such that for all we have for every .
Fix some . The polynomial has the form . After identifying the matrix with a vector in for , we see that is a sum of terms of the form
where the are aggregations of the for identical entries of , and are polynomials obtained from the entries of under . We can join the polynomials into a single polynomial , which would also absorb . Thus, we rewrite in the form where each is a polynomial with rational coefficients (as the coefficients in are rational).
In order to reason about the sign of this expression as , we need to find the leading term of . This, however, is easy: the exponents are algebraic numbers, and are therefore susceptible to effective comparison. Thus, we can order the terms by magnitude. Then, we can determine the asymptotic sign of each coefficient by looking at the leading term in .
We can thus determine the asymptotic behaviour of each , to conclude whether eventually holds. Moreover, for rational , every quantity above can be computed to arbitrary precision, therefore it is possible to compute a threshold , after which, for all , holds. This completes the proof.
For a semi-algebraic set , it is decidable whether there exists a o-minimal invariant, disjoint from , that contains the orbit after some time . Moreover in the positive instances an invariant that is -definable can be constructed.
If there is an invariant that contains , for some , then Section 3.2 implies that there exists some such that is contained in . Consequently, the question that we want to decide is equivalent to the question of whether there exists a , such that . The latter is decidable thanks to Section 4.1. The effective construction follows from the fact that such a is computable and that the cone is -definable.
4.2 Effectively Constructing the Semi-algebraic Invariant
We now turn to show that in fact, for semi-algebraic error sets , we can approximate with a semi-algebraic set such that if avoids , so does the approximation. Intuitively, this is done by relaxing the “non semi-algebraic” parts of in order to obtain a fat cone. This relaxation has two parts: one is to “rationalize” the (possibly irrational) exponents , and the other is to approximate the polylogs in by polynomials.
Relaxing the exponents.
We start by approximating the exponents with rational numbers. We remark that naively taking rational approximations is not sound, as the approximation must also adhere to the additive relationships of the exponents.
Let and be tuples of rational numbers such that for . Define as:
Thus, captures the integer additive relationships among the . Define
Let . We simply replace by such that . Note that it is not necessarily the case that , so this replacement is a-priori not sound. However, for large enough the inequalities do hold, which will suffice for our purposes.
We can now define the fat cone. Let and and as above, the fat orbit cone is the set:
That is, the fat cone is obtained from with the following changes:
is replaced with , where the are rational approximations of the , and maintain the additive relationships.
is replaced with where .
The variable starts from (as opposed to ).
We first show that the fat cone is semi-algebraic (the proof is in Appendix C), then proceed to prove that if there is a cone that avoids the error set, then there is a fat one that avoids it as well.  is definable in , and we can compute a representation of it. Let be a a semi-algebraic error set such that for some , then there exists as above such that
for every it holds that .
The result is constructive, so when is given, the constants can be computed. It follows that a corollary of this lemma, and Section 4.2, is a stronger statement than that of Section 4.1, namely one where is replaced by . We state it here before moving on with the proof of Section 4.2.
For a semi-algebraic set , it is decidable whether there exists a o-minimal invariant, disjoint from , that contains the orbit after some time . Moreover in the positive instances an invariant that is -definable can be constructed.
The proof of Section 4.2 is given by the two corresponding steps. The second step, proving the invariance of the fat cone, is Appendix C in Appendix C. We turn our attention to the first step. Let be a semi-algebraic error set, and let be such that , then there exists as above such that .
We use the same analysis and definitions of , , , as in the proof of Section 4.1 and focus on a single polynomial . Recall that we had
where each is a polynomial with rational coefficients.
Denote . We show, first, how to replace the exponents vector by any exponents vector in for appropriate , and second, how to replace by where for some appropriate and , while maintaining the inequality or equality prescribed by .
Denote by the set of vectors of exponents in (1). Let , such that for every , if then . That is, is a lower bound on the minimal difference between distinct exponents in (1). Observe that we can compute a description of , as the exponents are algebraic numbers.
Let (where is the Euclidean norm in ). Let be such that , then, for all , if then .
Proof of Section 4.2.
Suppose that , then by the above we have , and hence
We can now choose and such that and for all we have
It follows from Section 4.2 and from the definition of that, intuitively, every maintains the order of magnitude of the monomials in .
More precisely, let for some , then the exponent of the ratio of every two monomials in has the same (constant) sign as the corresponding exponent in . Moreover, the exponents of distinct monomials in differ by at least in .
We now turn our attention to the factor. First, let be large enough that has constant sign for every . We can now let be large enough such that for every , the sign of coincides with the sign of for every . It remains to give an upper bound on of the form such that plugging instead of does not change the ordering of the terms (by their magnitude) in .
Let be the maximum degree of all polynomials in (1), and define (in fact, any would suffice), then we have that, for , has the same sign as for every (by our choice of ), and guarantees that plugging instead of does not change the ordering of the terms (by their magnitude) in . Since the exponents of the monomials in differ by at least , it follows that their order is maintained when replacing by .
Let for some and , then by our choice of , the dominant term in is the same as that in . Therefore, for large enough , the signs of and are the same.
Note that since , then w.l.o.g. for every . Thus, by repeating the above argument for each , we can compute , , , and such that , and we are done.
5 A Reduction from Zeros of an Exponential Polynomial
In Section 4.2, we showed unconditional decidability for the question of whether there exists an invariant containing the orbit , for some . Even though we construct such an invariant, it cannot be used as a certificate proving that the orbit never enters the error set; however it is a certificate that the orbit of the system does not enter after time .
In this section we give indications that deciding whether there exists an invariant that takes into account the orbit is difficult. More precisely, we will reduce a problem about zeros of a certain exponential polynomial to the question of whether there exists a semi-algebraic invariant disjoint from containing .
In the setting of discrete linear dynamical systems, the existence of a semi-algebraic invariant from time immediately implies the existence of one from time . This is because the system goes through finitely many points from to , which can be added one by one to the semi-algebraic set. In this respect CDSs are more complicated to analyse.
The problem that we reduce from, can be stated as follows. We are given as input real algebraic numbers , and , and asked to decide whether the exponential function:
has any zeros in the interval . This is a special case of the so-called Continuous Skolem Problem [Bell10, chonev2016skolem].
While there has been progress on characterising the asymptotic distribution of complex zeros of such functions, less is known about the real zeros, and we lack any effective characterisation, see [Bell10, chonev2016skolem] and the references therein. The difficulty of knowing whether has a zero in the specified region is because (a) all the zeros have to be transcendental (a consequence of Hermite-Lindemann Theorem) and (b) there can be tangential zeros, that is has a zero but it never changes its sign. See the discussion in [Bell10, Section 6]. Finding the zeros of such a polynomial is a special case of the bounded continuous Skolem problem. We note that when are all rational the problem is equivalent to a sentence of (and hence decidable) by replacing .
The rest of this section is devoted to the proof of the following theorem.
For every exponential polynomial we can construct a CDS and semi-algebraic set such that the following two statements are equivalent:
there exists a semi-algebraic invariant disjoint from that contains ,
does not have a zero in .
Fix the function , i.e. real algebraic numbers and . Without loss of generality we can assume that are all nonnegative, since if and only if where is larger than all .
Since every is algebraic, there is a minimal polynomial , that has as a simple root. Let be the companion matrix of the polynomial . The numbers are eigenvalues of multiplicity one, and the latter also has zero as an eigenvalue of multiplicity two. In addition to those, the matrix generally has other (complex) eigenvalues as well. We put in Jordan normal form, where is made of two block diagonals: and , where
and is some matrix. Define:
the vector that has ones and the rest, zeros, whose purpose is to ignore the contribution of the eigenvalues in matrix in the system. To simplify notation, since is ignoring the contribution of the matrix , the dynamics of the system can be assume to be the same as:
Focus on a single eigenvalue, i.e. on the graph , as the analysis will easily generalise to the CDS in question. This is itself a CDS, so terminology such as orbits etc. make sense. The challenge is to find a family of tubes around this exponential curve such that (a) all the tubes together with are invariants and (b) the tubes are arbitrarily close approximations of the curve.
We achieve this by the following families of polynomials:
under-approximations are given by the family indexed by :
over-approximations are given by a family indexed by and :
It is clear from Taylor’s theorem and the assumption that , that by taking , and the sets are arbitrary precise approximations of the graph , what remains to show is that they are invariant.
 For every there exists such that for all