HUNGARIAN JOURNAL OF INDUSTRY AND CHEMISTRY Vol. 46(2) pp. 19-25 (2018) hjic.mk.uni-pannon.hu DOI: 10.1515/hjic-2018-0013 STRONG REACHABILITY OF REACTIONS WITH REVERSIBLE STEPS ESZTER VIRÁGH ∗1 AND BÁLINT KISS1 1Department of Control Engineering and Information Technology, Budapest University of Technology and Economics, Magyar Tudósok krt. 2, Budapest, 1117, HUNGARY The controllability of reactions is an important issue in the chemical industry. The control of reactions is of great practical interest in order to ensure the energy- and time-efficient production of compounds. This paper studies the dynamical models of some chemical reactions in order to verify their controllability with regard to a candidate input signal, namely the change in the ambient temperature of a reaction. Keywords: reversible reaction, strong reachability, controllability, Lie algebra 1. Introduction Chemical reactions are widely applied during the synthe- sis and transformation processes of organic compounds. The reaction mechanism and resulting products depend mainly on the concentrations of the species, the catalyst used, the ambient temperature, and the ambient pressure. If the values of these parameters are changed, one can obtain different products from the original ones but it is also possible to increase the productivity and energy- efficiency of the reaction. Hence the application of a proper feedback law to ensure the latter scenario may be envisaged. A study of the local controllability by considering the reaction rate coefficient as an input has been presented in Ref. [1]. The authors of Ref. [2] have also consid- ered the reaction rate coefficient as an input and extended the results by claiming that global controllability holds. The controllability of another control input, namely the dilution ratio, is studied in Ref. [3]. General conditions for strong reachability in the case of a temperature input were reported earlier in Ref. [4]. Moreover, the conditions of strong reachability for polymer electrolyte membrane fuel cells (PEMFC), controlled by concentrations, have also been analysed. The motivation to consider the con- centrations and temperature (or their rate of change) as input signals is due to the fact that these quantities can be easily modified efficiently by industrial equipment that is currently in use, thus these results can be used as a ba- sis to establish control laws to stabilize a desired reaction performance without major changes being made to the equipment used in production. The oxidation of acetone with hydroxylamine (the oximation reaction) was inves- tigated by Raman spectroscopy [5]. Knowing the mecha- ∗Correspondence: viragh.eszter@gmail.hu nism, the controllability is important for this reaction. Throughout this paper, the candidate variable for con- trol is the rate of change in the temperature Ṫ , i.e. the first time derivative of the ambient temperature. From a practical point of view, this is a simplification since the variable which can be changed externally, denoted by u, is not Ṫ but an algebraic expression involving u and other variables of the system as well. For the dynamics consid- ered in this paper it is always possible, however, to obtain the values of u as a function of Ṫ and other state vari- ables. The remaining part of this paper is organized as fol- lows. Sec. 2 briefly revisits the concepts related to the strong reachability of nonlinear dynamical systems and conditions of strong reachability. The differential equa- tions describing the dynamics of reactions are presented so that the rate of change in temperature is considered as the controlled input in Sec. 3. In Sec. 4 the strong reach- ability of the oximation reaction is studied. The systems in the case of acidic medium in Subsection 4.1 and in weakly basic medium in Subsection 4.2 are analyzed. In Ref. [4] a sufficient condition for strong reachabil- ity was given for reactions with general dynamics. In Sec. 5, the conditions for strong reachability are given, if the reaction also contains reversible steps. In the last section the conclusions of the paper are drawn. 2. Study of strong reachability Consider the following nonlinear dynamical system, given by the differential equation: ξ̇ = f(ξ) + g(ξ)u, ξ(0) = ξ∗ ∈ Rn, (1) where f,g ∈ C∞(Rn,Rn) are smooth vector fields and u ∈ R is the control-input variable. The vector fields mailto:viragh.eszter@gmail.hu 20 VIRÁGH AND KISS f and g are often referred to as drift and control vector fields, respectively. For the sake of completeness, let us revisit some definitions used in Refs. [4, 6]. Definition 1 (Reachability set). Consider the system given by Eq. 1. The set R(ξ∗, t) ⊂ Rn is referred to as the reachability set from the point ξ∗ at time t and it is the union of values at t of the solutions to Eq. 1 for some admissible input function u with the initial condi- tion ξ(0) = ξ∗. Definition 2 (Strong reachability). The system Eq. 1 is referred to as strongly reachable from the point ξ∗, if the set R(ξ∗, t) has an interior point for all t > 0. Definition 3 (Lie bracket). Suppose that f ∈ C∞(Rn,Rn) and g ∈ C∞(Rn,Rn), then the Lie bracket of the vector fields f and g is [f,g] = Dg f −Df g. (2) The operator adngf : C ∞(Rn,Rn) × C∞(Rn,Rn) → C∞(Rn,Rn) is defined as: ad0gf = f, ad n gf = [g, ad n−1 g f]. (3) Definition 4 (Lie algebra). Consider the vector fields f,g ∈ C∞(Rn,Rn). The Lie algebra generated by f and g is denoted by Λ = Lie(f,g) and is the smallest lin- ear subspace of C∞(Rn,Rn) that satisfies the following conditions: 1. f, g ∈ Λ , 2. for any a,b ∈ Λ, [a,b] ∈ Λ. It should be noted that Λ also defines a distribution. Definition 5 (Distribution). The distribution ∆ is the op- erator which assigns a linear subspace of RN to ∀x ∈ RN . Definition 6 (Controllability distribution). The controlla- bility distribution ∆c of Eq. 1 is the smallest distribution which satisfies the following conditions: 1. g ∈ ∆c, 2. ∆c is invariant to the vector field f (∀η ∈ ∆c, [η,f] ∈ ∆c), 3. ∆c is involutive (∀η1,η2 ∈ ∆c, [η1,η2] ∈ ∆c). The controllability distribution has a subspace spanned by vector fields g and [f,g]. The following theo- rem is a fundamental result used in Ref. [6]. Theorem 1 (Reachability rank condition). Consider the controllability distribution ∆c of Eq. 1. The sys- tem Eq. 1 is strongly reachable at point ξ∗ ∈ Rn if dim{∆}c (ξ ∗) = n. 3. Strong reachability of kinetic equations The active control of chemical processes may be nec- essary to maximize the amount of target products and minimize the amount of by-products. To achieve such a control objective, a suitable input variable must be se- lected so that the resulting dynamical system is control- lable from that input. To check if this requirement is sat- isfied, the tools introduced in the previous section will be applied to the equations describing the reaction dynam- ics. Consider a system of R reaction steps and with M species (R,M > 0). By borrowing notational conven- tions from chemistry, each reaction step can be generally defined by M∑ m=1 α(m, r)X(m) kr−−→ M∑ m=1 β(m, r)X(m), r = 1, 2, . . . ,R, (4) where α(·, r) = (α(1, r),α(2, r), . . . ,α(m, r))T denotes the reactant complex vector, β(·, r) = (β(1, r),β(2, r), . . . ,β(m, r))T represents the product complex vector, X(m) is the mth species and kr is the re- action rate coefficient of the rth reaction step. The species on the left-hand side of Eq. 4 are referred to as reactant species and reactant complexes refer to their formal lin- ear combinations. Similarly, one may find the products and their linear combinations (product complexes) on the right-hand side of the reaction described in Eq. 4. Let us also define the stoichiometric matrix, denoted by γ. The matrix γ consists of R columns and M rows, such that each column is obtained by γ(·, r) = β(·, r) −α(·, r). (5) Eq. 4 defines the reaction but it does not specify its mass action kinetics. However, in order to study the controlla- bility, the differential equations of the reaction dynamics need to be obtained in the form of differential equation Eq. 1. These equations are obtained from the heat bal- ance [7–9] of reaction Eq. 4 as ẋm = R∑ r=1 γ(m, r)krx α(·, r) m = 1, 2, . . . ,M, (6) Ṫ = R∑ r=1 1 βr,0 krx α(·, r) + u, (7) where xm denotes the concentration of species m, T is the temperature, and xα(·, r) = ∏M p=1 x α(p, r) p . Re- call that the single input u appears in the expression of Ṫ . The state vector ξ for the dynamics Eqs. 6–7 reads ξ = (x1,x2, . . . ,xm,T) T . The reaction rate coefficient kr can be given as kr = kr,0e − Er R0T (8) Hungarian Journal of Industry and Chemistry STRONG REACHABILITY OF REACTIONS WITH REVERSIBLE STEPS 21 where kr,0,Er,R0 ∈ R+. To study reachability, one has to determine the num- ber of linearly independent Lie brackets spanning the Lie algebra generated by the vector fields g and adgf. Thanks to the special structure of the reaction dynamics, the lin- ear independence can be examined by factorizing the ma- trix of Lie brackets and checking the rank of the factors. First, let us define the reaction dynamics matrix: Definition 7 (Reaction dynamics matrix DR). Introduce the notation k (i) j := ∂i ∂Ti kj i,j ∈{1, 2, . . . ,R} (9) The matrix of size R×R of the derivatives of the reaction rate coefficient defined as DR =   k (1) 1 k (2) 1 . . . k (R) 1 k (1) 2 k (2) 2 . . . k (R) 2 ... ... . . . ... k (1) R k (2) R . . . k (R) R   (10) is referred to as the reaction dynamics matrix. The following Lemma and Theorem have been shown in Ref. [4]. They are also provided here for completeness. Lemma 1. Consider a reaction with R steps. Suppose that the activation energies E1,E2, . . . ,ER of the reac- tion steps are all different and strictly positive, then the reaction dynamics matrix DR is of full rank for every T > 0. Theorem 1 cannot be applied directly to the reaction dynamics Eq. 6–7, because the right-hand sides (RHS) of some equations in Eq. 6 may be linearly dependent. Let δ denote the number of linearly dependent RHSs in Eq. 6. The system of chemical reactions is considered to be strongly reachable from a point ξ∗ ∈ RM+1 if and only if the dimension of the controllability distribution is M −δ + 1. The additional dimension is due to Eq. 7 with the temperature T as an additional variable. Using Theorem 1, the controllability subspace for par- ticular reactions can be deduced, so the conditions for strong reachability of the reactions can be determined. Theorem 2. Consider a reaction with M species and R reaction steps. Suppose that the activation energies E1,E2, . . . ,ER of the reaction steps are all different and strictly positive. Then the reaction dynamics with the tem- perature change (Ṫ ) as an input variable are strongly reachable if the concentrations of all reactant species are positive. Theorem 2 provides a condition for the strong reach- ability of reactions in a general form. However, for some reactions where the reaction dynamics have additional properties, weaker conditions may be sufficient to ensure strong reachability. In Sec. 4, the strong reachability con- ditions in the case of oximation reactions were investi- gated. 4. Controllability study of the oximation re- action The oxidation of acetone with hydroxylamine was inves- tigated by Raman spectroscopy in Ref. [5]. The reaction is strongly exothermic and the concentration of the inter- mediate highly depends on the pH and temperature. The process can be hazardous, however, it is not dangerous to run in a laboratory with low concentrations and in a con- trolled manner. Strong reachability is a necessary condi- tion to be able to control the reaction. 4.1 Oximation reaction in acidic medium In oximation reactions, the sequence (number and nature) of reaction steps depends on the pH. The equations of the reaction steps are different in acidic and weakly basic media. In the case of acidic media the reaction takes place over two reaction steps as given by For the sake of notational simplicity, the symbols A, B, C, D, E, and F will denote the species such that the two reaction steps above read: A + B k1−−→ C (11) C + D k2−−→ E + F. (12) Let us denote the concentrations of the species by a,b,c,d,e,f ≥ 0. It is assumed that the reaction rate coefficients k1,k2 > 0. Theorem 2 implies that the reaction is strongly reach- able, provided that the conditions are met. It follows from strong reachability that it is possible to arrive at any con- centrations of M − δ species and at any temperatures by suitable manipulation of the input. Recall that there is no guarantee that such concentrations and temperatures also define a steady-state for the system. Note that Theorem 2 only provides a sufficient condi- tion for strong reachability. For chemical reactions, the positivity condition of activation energies is almost al- ways satisfied. Considering the reaction steps of the ox- imation reaction in acidic media the two remaining con- ditions of strong reachability will be studied: (1) the pos- itivity of the concentration of all reactant species; (2) the distinctness of activation energies. Let us now suppose that the system in Eqs. 11–12 is strongly reachable. The stoichiometric matrix for the re- 46(2) pp. 19-25 (2018) 22 VIRÁGH AND KISS action steps reads: γ =   −1 0 −1 0 1 −1 0 −1 0 1 0 1   (13) and it is easy to verify that δ = 2 in this case. The differ- ential equation of the reaction:  ȧ ḃ ċ ḋ ė ḟ Ṫ   =   −k1ab −k1ab k1ab−k2cd −k2cd k2cd k2cd k1 β ab + k2 β cd   +   0 0 0 0 0 0 1   u = = f(ξ) + g(ξ)u (14) is in a form similar to Eq. 1, where ξ = (a,b,c,d,e,f,T)T and the vector field g(ξ) is constant. Since the rank of the stoichiometric matrix γ is 2 and the temperature is a scalar quantity, Theorem 1 im- plies that the system is strongly reachable if dim{∆}c = 2 + 1 = 3. Hence, to study strong reachability, the num- ber of linearly independent vector fields spanning the Lie algebra generated by the vector fields adgf and g must be determined. The Lie brackets adgf and ad 2 gf read: ad(i)g f =   −k(i)1 ab −k(i)1 ab k (i) 1 ab−k (i) 2 cd −k(i)2 cd k (i) 2 cd k (i) 2 cd k (i) 1 β ab + k (i) 2 β cd   , (15) where i ∈{1, 2}. To study the dimension of the controllability distribu- tion ∆c one has to determine the rank of the matrix whose columns are g, adgf and ad 2 gf which reads:( adgf ad 2 gf g ) = =   −k(1)1 ab −k (2) 1 ab 0 −k(1)1 ab −k (2) 1 ab 0 k (1) 1 ab−k (1) 2 cd k (2) 1 ab−k (2) 2 cd 0 −k(1)2 cd −k (2) 2 cd 0 k (1) 2 cd k (2) 2 cd 0 k (1) 2 cd k (2) 2 cd 0 k (1) 1 β ab + k (1) 2 β cd k (2) 1 β ab + k (2) 2 β cd 1   . (16) The last row is linearly independent of all other rows in the matrix of Eq. 16, hence, by deleting the last row and column from the matix, the rank will be decreased by 1. The remaining matrix is denoted by Θ and defined as Θ =   −k(1)1 ab −k (2) 1 ab −k(1)1 ab −k (2) 1 ab k (1) 1 ab−k (1) 2 cd k (2) 1 ab−k (2) 2 cd −k(1)2 cd −k (2) 2 cd k (1) 2 cd k (2) 2 cd k (1) 2 cd k (2) 2 cd   . (17) It is clear that the condition dim{∆c} = 3 holds true if and only if rank(Θ) = 2. It is easy to see that the matrix Θ can be factorized as Θ =   −ab 0 −ab 0 ab −cd 0 −cd 0 cd 0 cd   ( k (1) 1 k (2) 1 k (1) 2 k (2) 2 ) = A ·D2. (18) The condition of rank(Θ) = 2 can hold true if and only if the matrices A and D2 are of full rank according to the multiplication theorem of determinants. Matrix A is of full rank (rank(A) = 2) if and only if a 6= 0, b 6= 0, c 6= 0 and d 6= 0. The reaction dynamics matrix D2 is of full rank if and only if there is no constant c ∈ R \ {0} such that k(2)1 = ck (1) 1 and k (2) 2 = ck (1) 2 , hence k (2) 1 k (1) 1 = k (2) 2 k (1) 2 (= c) (19) cannot be true. Recalling that the reaction rate coeffi- cients are kr = kr,0e − Er R0T (kr,0, Er, and R0 are positive constants), it is easy to determine the time derivatives: k(1)r = ( Er R0T2 ) kr,0e −Er R0T , (20) k(2)r = kr,0 ( E2r R20 T 4 − 2Er R0 T3 ) e −Er R0T . (21) The ratios of the first- and second-order time derivatives are obtained as k (2) r k (1) r = kr,0 ( E2r R20 T 4 − 2ErR0 T3 ) e −Er R0T( Er R0T2 ) kr,0e −Er R0T = Er − 2R0T R0T2 . (22) Based on Eq. 22, the equality in Eq. 19 holds true if and only if E1 = E2. As a result it has been proven that if the reaction dynamics are strongly reachable then the concentrations a, b, c and d are positive and E1 6= E2. Thus the condi- tions of Theorem 2 are also necessary for strong reacha- bility in the case of oximation reactions in acidic media. 4.2 Oximation reaction in weakly basic medium In the case of weakly basic media the reaction occurs ac- cording to the reaction steps given by: Hungarian Journal of Industry and Chemistry STRONG REACHABILITY OF REACTIONS WITH REVERSIBLE STEPS 23 Since the specific chemical compositions of the species are irrelevant to the controllability analysis, the symbols A, B, C, D, E, F, and G will denote the species such that the above reaction steps read: A + B k1−−→ C, (23) C k2−−→ D + E, (24) D + F k3−−⇀↽−−− k−3 G + E. (25) Let us denote the concentration of the species by a, b, c, d, e, f, g ≥ 0. It is assumed that the reaction rate coeffi- cients are strictly positive: k1, k2, k3, k−3 > 0. Recall that Theorem 2 only provides a sufficient con- dition for strong reachability. By considering oximation reactions in weakly basic media the remaining conditions of strong reachability will be studied. Let us suppose now that the system of Eqs. 23–25 is strongly reachable. The stoichiometric matrix for the re- action steps reads: γ =   −1 0 0 0 −1 0 0 0 1 −1 0 0 0 1 −1 1 0 1 1 −1 0 0 −1 1 0 0 1 −1   . (26) The differential equation of the reaction  ȧ ḃ ċ ḋ ė ḟ ġ Ṫ   =   −k1ab −k1ab k1ab−k2c k2c−k3df + k−3ge k2c + k3df −k−3ge −k3df + k−3ge k3df −k−3ge k1 β ab + k2 β c + k3 β df + k−3 β ge   + +   0 0 0 0 0 0 0 1   u = f(ξ) + g(ξ)u (27) is in a form similar to Eq. 1, where the vector field g(ξ) is constant and ξ = (a,b,c,d,e,f,g,T)T . Since the rank of the stoichiometric matrix γ is 3 and the temperature is a scalar quantity, Theorem 1 implies that the system is strongly reachable if dim{∆}c = 3 + 1 = 4. Hence, the number of linearly independent vector fields spanning the Lie algebra generated by the vector fields adgf and g must be determined. The Lie–brackets adgf, ad 2 gf and ad 3 gf read: adigf =   −k(i)1 ab −k(i)1 ab k (i) 1 ab−k (i) 2 c k (i) 2 c−k (i) 3 df + k (i) −3ge k (i) 2 c + k (i) 3 df −k (i) −3ge −k(i)3 df + k (i) −3ge k (i) 3 df −k (i) −3ge k (i) 1 β ab + k (i) 2 β c + k (i) 3 β df + k (i) −3 β ge   , (28) where i ∈ {1, 2, 3}. To study the dimensions of the controllability distribution ∆c one has to give the rank of the matrix whose columns are adgf, ad 2 gf, ad 3 gf, and g:( adgf ad 2 gf ad 3 gf g ) . (29) The last row is linearly independent of all other rows in matrix Eq. 30, hence, by deleting the last row and column from the matrix, the rank will be decreased by 1. The remaining matrix is denoted by Θ and defined as Θ =     −k(i)1 ab −k(i)1 ab k (i) 1 ab−k (i) 2 c k (i) 2 c−k (i) 3 df + k (i) −3ge k (i) 2 c + k (i) 3 df −k (i) −3ge −k(i)3 df + k (i) −3ge k (i) 3 df −k (i) −3ge   i=1,2,3   (30) The condition dim{∆}c = 4 holds true if and only if rank(Θ) = 3. It is easy to see that the matrix Θ can be factorized as Θ = A ·D =  −ab 0 0 0 −ab 0 0 0 ab −c 0 0 0 c −df ge 0 c df −ge 0 0 −df ge 0 0 df −ge     k (1) 1 k (2) 1 k (3) 1 k (1) 2 k (2) 2 k (3) 2 k (1) 3 k (2) 3 k (3) 3 k (1) −3 k (2) −3 k (3) −3   (31) The condition rank(Θ) = 3 can hold true only if rank(A) ≥ 3 and rank(D) ≥ 3. T he 3rd and 4th columns in matrix A are linearly dependent, thus rank(A) ≤ 3. The condition rank(A) = 3 can hold true only if the concentrations a, b, and c as well as the 46(2) pp. 19-25 (2018) 24 VIRÁGH AND KISS concentrations d and f, or the concentrations g and e are strictly positive. Matrix D is of full rank only if D con- sists of a 3 × 3 times full-rank matrix. Lemma 1 implies that the reaction dynamics matrix D3 is of full rank if the activation energies are different. Hence, matrix D is of full rank, if 3 different activation energies exist. The system was proven to be strongly reachable if 3 of the activation energies are all different and concentrations a, b, c and d, f, or g, e are positive. Thus a condition for strong reachability coule be given more precisely in the case of oximation reaction in weakly basic media. 5. Reversible reaction step For reactions of general types, Theorem 2 provides a con- dition for strong reachability. However, it will be shown that if the reaction contains one or more reversible steps, weaker conditions are sufficient for strong reachability. The concentrations are defined by x1, x2, . . . , xM , as in the previous sections. The notation A is introduced for the matrix describing the effect of concentrations: A := γ diag(xα(· ,1),xα(·, 2), . . . ,xα(·, R)), (32) where xα(·,r) = ∏M m=1 x α(m, r) m and γ is the stoichio- metric matrix as introduced by Eq. 5. The vector k = (k1,k2, . . . ,kR) T is composed of the reaction rate coef- ficients. The notation D is introduced for the following matrix composed of the derivatives of reaction rate coef- ficients: D := ( k(1) k(2) ... k(rank(γ)) ) . (33) Lemma 2. The reaction dynamics in Eqs. 6–7 with the input variable Ṫ are strongly reachable, if rankγ = rank(AD), where γ is the stoichiometric matrix and A and D are defined as above. Proof The differential equation of the reaction reads: ( ẋ Ṫ ) =   vR∑ r=1 kr βr,0 xα(·, r)   + ( 0 1 ) u, (34) where ẋ = (ẋ1, ẋ2, . . . , ẋM )T , u is the control input and the vector field v stands for the vector composed of the right-hand sides of Eq. 6. As in the previous sections, the study of strong reach- ability means verification of the dimension of the control- lability distribution ∆c. The dimension of the controlled input (the dimension of the change in temperature) is 1, thus, Theorem 1 implies that the system is strongly reach- able if and only if dim{∆}c = rank(γ) + 1. The vector fields spanning the controllability distri- bution are g and adigf for i > 0. The Lie bracket ad i gf reads: adigf =   v (i) R∑ r=1 k (i) r βr,0 xα(·, r)  =   A·k (i) R∑ r=1 k (i) r βr,0 xα(·, r)   (35) for i ∈{1, 2, . . . , rankγ}. The rank of the controllability distribution is hence the rank of the matrix( adgf ad 2 gf . . . ad rankγ g f g ) =  A·k (1) . . . A·k(rankγ) 0 R∑ r=1 k (1) r βr,0 xα(·, r) . . . R∑ r=1 k(rankγ) βr,0 xα(·, r) 1   (36) The last row in Eq. 36 is linearly independent of the oth- ers, hence, the same reasoning as earlier is followed and the last row and columns are eliminated, thus, decreasing the rank by one. The resulting matrix is denoted by Θ and reads: Θ = ( A·k(1) A·k(2) . . . A·k(rankγ) ) = = AD. (37) Since the dimension of the controllability distribution is rank(Θ) + 1, the reaction dynamics are strongly reach- able if rankγ = rankΘ or if rankγ = rank(AD). Theorem 3. Consider the reaction dynamics Eqs. 6–7 such that the activation energies E1, E2, . . . , ER are positive and different in pairs. Suppose that the concen- trations of reactant species are positive in the case of one- way reaction steps and at least one of the ways is positive in the case of reversible reaction steps. Then, the reaction dynamics controlled by Ṫ are strongly reachable. Proof If the system does not contain reversible reaction steps, Theorem 2 is obtained. Without loss of generality, it can be supposed that the system contains one reversible reaction step. This step can be replaced by pairs of irreversible reaction steps, with reaction rate coefficients denoted by ke and k−e. The changes in the concentrations are equal in the reac- tion step with rate coefficient ke and in the reaction step with rate coefficient k−e, only the direction is different. Thus, the two columns in matrix γ for the reversible re- action steps are always linearly dependent. Lemma 2 im- plies that the system is strongly reachable if and only if rankγ = rank(AD). If the activation energies are pos- itive and all different, Lemma 1 implies that matrix D is of full rank. The matrix A is defined by Eq. 32, thus, the columns for ke and k−e are linearly dependent. The column for ke contains the factor of the concentrations of the reactant species in the transformation step and k−e contains the factor of the concentrations of the re- actant species in the transformation step in the opposite direction with the arbitrary sign in the place of non-zero elements. By substituting one of the two vector fields with a zero vector field, the rank of matrix A remains unchanged. Thus, in the case of reversible reactions for strong reachability, it is sufficient if the reactant species have positive concentrations only in one of the directions, and the activation energies are positive and all different. Hungarian Journal of Industry and Chemistry STRONG REACHABILITY OF REACTIONS WITH REVERSIBLE STEPS 25 6. Conclusion The reaction dynamics of strong reachability where the control variable is selected as the rate of change in the ambient temperature (Ṫ) have been studied. First, the strong reachability was analyzed in the case of the oxi- mation reaction. Since the processes depend on the pH, conditions that facilitate strong reachability in acidic as well as weakly basic media were studied. For our anal- ysis, Theorem 2 was used. It provides sufficient condi- tions to facilitate strong reachability, however, these con- ditions are not always necessary. It was proven that the conditions in Theorem 2 are necessary to facilitate strong reachability of the oximation reaction in the case of acidic media. In weakly basic media, the system contained a re- versible reaction step, thus, the conditions of Theorem 2 could be determined. Strong reachability has also been studied for reaction dynamics of a general type that contain at least one re- versible reaction step where the conditions of Theorem 2 could also be further refined. For reversible reaction steps it has been shown that positive reactant concentrations are unnecessary to facilitate strong reachability in both direc- tions of the reversible steps, in one direction is sufficient. 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