Finance Minister Ahmed Shide said on Thursday that the government was trying to make people in the city aware of the military operation.
He said on Saturday evening federal troops had taken control of the Tigrayan capital Mekelle within hours of launching an offensive there, laying to rest fears of protracted fighting in the city of 500,000 people. At least one rocket fired from Tigray targeted Eritrea's capital Asmara on Friday night, regional diplomats said. That makes it hard to verify claims about the fighting, but humanitarians have said at least hundreds of people have been killed. "This is about defending our right to self-determination".
The African Union envoys were in Addis Ababa "with a view to help mediate between the parties to the conflict" in Ethiopia, South African President Cyril Ramaphosa, who is also the AU chair, said earlier this week.
Mr Abiy's statement said federal forces would "continue their task of apprehending TPLF criminals and bring them to the court of law". They characterise the conflict as a "law and order operation", which they predict will end rapidly.
Tigrayan officers, civil servants and others reportedly are being ethnically profiled outside of Tigray, and many echo the Tigrayan leaders' claim that this is an attack on the entire group, a sentiment that may seed support for secession.
He's given the state's ruling Tigray People's Liberation Front until Wednesday evening to surrender, failing which he's warned that his forces will attack their stronghold of Mekelle.
According to Ahmed, the ENDF has thus far managed to secure the release of thousands of Northern Command officers held hostage by the TPLF, secure the Northern Command camp, and take control of the airport, public institutions, the regional administration office and other critical facilities.
It was not immediately clear how close the army was to the city.
The TPLF said the central government's mandate to rule had expired, arguing that Mr Abiy had not been tested in a national election.
"The Prime Minister discussed at length the patience with which his government handled the provocations and destabilization agenda the TPLF orchestrated for more than two years", Abiy's office said. Abiy's government declared the Tigray election illegal.
The minister dismissed the idea that Ethiopia has been left with "any, like, severe wound" from the conflict, and he expressed confidence in the prime minister's ability to restore normalcy.
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One Ethiopian military official claimed that more than 10,000 "junta forces" have been "destroyed" since the fighting began on November 4, when Abiy accused the TPLF of attacking a military base. Col.
A separate TPLF statement, reported by AFP, urged "the worldwide community to condemn the artillery and warplane attacks and massacres being committed".
Abiy did not mention in his statements whether there had been casualties in the offensive to capture Mekelle.
The TPLF which once dominated Ethiopia's government have been sidelined under Mr Abiy's - last year's Nobel Peace Prize victor - rule.
Refugees have also told The Associated Press that Ethiopian forces near the border with Sudan are impeding people from leaving Ethiopia, while refugee crossings have largely slowed to a trickle.
For almost three decades, the front had controlled the Ethiopian political and security apparatus, before Abyei gradually removed it from power.
Tigray also borders the nation of Eritrea and the conflict has stirred concern about an escalation around the country of 115 million people, or in the volatile Horn of Africa region.
The Eritrean government has not responded to calls from Reuters for more than two weeks.
United Nations and western diplomats voiced deep concerns about the humanitarian situation in the restive region and called for unfettered access to the aid organisations and groups. But Eritrea and Abiy have warm relations.
It has not been possible to contact the Eritrean government for comment on this.